Logarithms | ICSE Class 9 Maths Notes

This note covers the meaning of logarithms, valid bases and arguments, conversion between exponential and logarithmic forms, connections with indices, the product, quotient and power laws, expansion of expressions, and combining logarithms.

What does a logarithm mean?

A logarithm tells us the exponent needed to obtain a given positive number from a chosen base. An exponent, also called an index, indicates the power to which the base is raised. The base is the number being raised to that power.

Definition: Let b be a positive real number other than 1, N a positive real number, and x a real exponent. Then $\log_b N=x$ means exactly that $b^x=N$. A real number is a number represented on the number line.

Read $\log_b N$ as “the logarithm of N to base b”. Here N is the argument, meaning the number whose logarithm is taken. The logarithm's value is x. The argument and the value have different roles, even though both are numbers.

Which restrictions belong to the definition?

The conditions are $b>0$, $b\ne1$, and $N>0$. The sign > means “greater than”, and ≠ means “not equal to”. A positive number is greater than zero. These conditions must be checked before manipulating a real logarithm.

The base cannot be 1: every real power of 1 is 1, so the exponent cannot be uniquely recovered. Positive bases give positive real powers. Zero and negative arguments therefore have no real logarithms under this definition.

The argument must be positive, but the logarithm's value need not be positive. A zero or negative exponent is still an exponent. Do not transfer the restriction on the argument to the answer when reading a logarithmic statement.

How should the notation be read?

In $\log_2 8=3$, the base is 2, the argument is 8, and the logarithm is 3. The statement asks which power of 2 gives 8. Since 2³ = 8, the required exponent is 3.

A common logarithm has base 10. Keep bases explicit when comparing expressions. Where a base is suppressed later in this note, all logarithms in that calculation share one fixed valid base. Suppressing the base does not remove the need for it.

How do you interchange exponential and logarithmic forms?

Exponential form displays the power directly: $b^x=N$. Logarithmic form makes the exponent the subject: $\log_b N=x$. To make a quantity the subject means to write it alone on one side of an equation. These forms express the same relationship.

The base stays the base in both forms. The result of raising the base to a power becomes the logarithm's argument. The exponent becomes the value of the logarithm. Nothing is multiplied or divided merely because the notation changes.

What steps keep the three roles clear?

1. Identify the base before moving any symbols.
2. Identify the exponent and the resulting positive number.
3. Write the alternative form, preserving all three roles.
4. Read the new statement back as a power to check it.

Worked example 1. Write 2³ = 8 in logarithmic form.

Answer: $\log_2 8=3$. The base remains 2; the result 8 becomes the argument; the exponent 3 becomes the logarithm. Checking the answer means returning to the given equality 2³ = 8.

Worked example 2. Write $\log_{10}10000=4$ in exponential form.

Answer: 10⁴ = 10000. The logarithm has value 4, so 4 must be the exponent of the base 10. The argument 10000 is the result of taking that power, not the exponent.

Worked example 3. Use 625 = 5⁴ to write a logarithmic statement.

Answer: $\log_5 625=4$. First read the given equality as 5⁴ = 625. Reversing the sides of an equality does not change its meaning. The base is 5 and the exponent is 4.

Worked example 4. Use 625 = 25² to write a logarithmic statement and compare it with the preceding result.

Answer: $\log_{25}625=2$. The same argument 625 has logarithm 2 to base 25 and logarithm 4 to base 5. Both are correct because the bases differ. A logarithm's value depends on its base as well as its argument.

Conversion is an exact restatement. It does not require a decimal approximation or a logarithm table when the power relationship is already known. Keep the given equality visible until the conversion has been checked.

How do the laws of indices explain logarithms?

Indices and logarithms describe the same powers from different directions. Indices tell us how powers combine. Logarithms record the exponents in those powers. The laws of logarithms can therefore be understood through multiplication, division and repeated powers with a common base.

Let m and n be real exponents, and retain the positive base b other than 1. For powers of this base, multiplication adds exponents, division subtracts exponents, and raising a power to another power multiplies exponents.

In this notation, adjacent letters or expressions indicate multiplication. The slash indicates division. Brackets identify an expression that must be treated as a whole. Thus $(b^m)^n$ means that the entire power $b^m$ is raised to n.

Why do zero and negative logarithms make sense?

The zero index gives $b^0=1$. In logarithmic form, this is $\log_b1=0$. It says that the logarithm of the argument 1 is zero; it does not say that zero can be used as an argument.

A negative index gives a reciprocal: $b^{-m}=1/b^m$. The reciprocal of a non-zero number is 1 divided by that number. This relationship allows a positive argument to have a negative logarithm without violating the argument condition.

For bases greater than 1, positive powers give numbers greater than 1, while negative powers give positive numbers less than 1. The dividing case is the zero power, which gives 1. These sign statements are being made for bases greater than 1.

Use the index law as a check. If a proposed logarithm law would make multiplication of powers multiply their exponents, something has gone wrong. Multiplication of powers with a common base adds their exponents.

How does the product law work?

A product is the result of multiplication; its factors are the quantities being multiplied. Let p and q denote positive real numbers. With the same valid base b throughout, the logarithm of their product is the sum of their logarithms.

Property: logarithm of a product

Definition: The product law is $\log_b(pq)=\log_b p+\log_b q$, where p and q are positive and b is positive but not 1. Both logarithms on the right use the base on the left.

To understand the law, let $m=\log_b p$ and $n=\log_b q$. By the definition of a logarithm, $p=b^m$ and $q=b^n$. Their product is therefore $pq=b^m b^n=b^{m+n}$.

The exponent producing pq is m + n. Consequently, $\log_b(pq)=m+n$, which is $\log_b p+\log_b q$. This reasoning explains the addition sign: it comes from adding the exponents when multiplying powers of the same base.

What does expansion by the product law mean?

Expansion means rewriting a logarithm of a combined expression as a sum or difference of simpler logarithms, with appropriate multipliers. In the product law, the combined expression is pq. Each factor becomes the argument of its own logarithm.

Worked example 5. Expand $\log_b(pq)$, given p and q positive, with b a positive base other than 1.

Answer: $\log_b(pq)=\log_b p+\log_b q$. The product of the 2 positive factors becomes a sum of their logarithms. Keep the same base in both terms; do not multiply the logarithm values.

The operation inside the logarithm determines whether this law applies. The law concerns multiplication of arguments. An addition sign between arguments does not become addition of their logarithms. In particular, the product law gives no permission to split a logarithm of a sum.

The positivity conditions also matter. A positive product alone does not establish that each separate logarithm exists. Before expanding into individual logarithms, check that each individual factor is positive, as required by the law.

How does the quotient law work?

A quotient is the result of division. In the fraction p/q, p is the numerator, the quantity being divided, and q is the denominator, the quantity by which it is divided. Here p and q are positive real numbers.

Property: logarithm of a quotient

Definition: The quotient law is $\log_b(p/q)=\log_b p-\log_b q$. The base b is positive and different from 1, and both p and q must be positive.

The order of subtraction follows the order of division. The numerator contributes the first logarithm. The denominator contributes the logarithm being subtracted. Changing that order changes the expression, except in cases where the resulting values happen to agree.

To derive the rule, set $m=\log_b p$ and $n=\log_b q$. Then $p=b^m$ and $q=b^n$, so $p/q=b^{m-n}$. The logarithm of the quotient is m − n, by the definition of a logarithm.

How do you preserve the subtraction sign?

Worked example 6. Expand $\log_b(p/q)$, given p and q positive, with b a positive base other than 1.

Answer: $\log_b(p/q)=\log_b p-\log_b q$. The result contains 2 logarithms. The denominator q supplies the subtracted term. Dividing the arguments does not mean dividing the values of their logarithms.

If a denominator contains a product, its entire logarithm is subtracted before that logarithm is expanded. Brackets help preserve this meaning. Removing brackets after a minus sign requires changing the sign of every term inside them.

A reciprocal follows the same rule. Since $\log_b1=0$, applying the quotient law gives $\log_b(1/p)=-\log_b p$. This is a consequence of the quotient law, rather than an unrelated formula to memorise.

As with the product law, a positive combined argument is not enough for every possible expansion. The separate numerator and denominator logarithms require positive arguments individually. Write the restrictions alongside the working whenever letters stand for unspecified numbers.

How does the power law turn an exponent into a multiplier?

A power is an expression formed by raising a base to an exponent. In $p^n$, p is the base of that power and n is its exponent. In $\log_b(p^n)$, b is the logarithm base; these two roles must be distinguished.

Property: logarithm of a power

Definition: The power law is $\log_b(p^n)=n\log_b p$, where p is positive, n is a real exponent, and b is positive but not 1. The exponent becomes a multiplier of the whole logarithm.

A coefficient is a number or expression multiplying another expression. On the right of the power law, n is the coefficient of $\log_b p$. It multiplies the logarithm's value. It does not multiply p inside the logarithm.

For a positive integer exponent, the power law follows by repeatedly applying the product law. An integer is a whole number, its negative, or zero. A positive integer exponent represents repeated multiplication of the positive argument by itself.

How does the squared case show the reasoning?

Worked example 7. Expand $\log_b(p^2)$, where p is positive and b is a positive base other than 1.

Answer: $\log_b(p^2)=\log_b(pp)=\log_b p+\log_b p=2\log_b p$. The square supplies two equal factors. Their logarithms are equal, so adding them gives twice the logarithm of p.

For a general exponent n, set $m=\log_b p$. Then $p=b^m$, and the index law gives $p^n=b^{mn}$. Its logarithm is mn, or $n\log_b p$. This explanation uses the corresponding law of real powers.

The law also holds for real exponents, including negative and fractional exponents, when the stated conditions hold. A fractional exponent expresses a power involving a root: a number whose specified positive integer power produces the original number. The positive square root has square equal to that number.

Keep brackets clear: $\log_b(p^n)$ and $(\log_b p)^n$ are different expressions. The first takes a logarithm after raising the argument to a power. The second raises the logarithm's value to a power. The power law applies to the first.

The requirement p > 0 belongs to this form of the rule. The fact that an even power can be positive does not allow a negative p to appear in the separate real logarithm on the right.

How do you expand an expression using several logarithm laws?

Let a, b and c now denote positive real quantities in an algebraic expression, and let y denote the value of that expression. In this section, b is a factor, not the logarithm base. All logarithms use one fixed positive base other than 1.

Consider $y=a^4b^2/c^3$. The numerator is the product of a⁴ and b²; the denominator is c³. The powers 4, 2 and 3 tell us the eventual coefficients. Division by c³ determines the negative sign of its logarithm term.

Which law should be used first?

The outermost operation is the operation combining the largest parts of the expression. Here it is division of the whole numerator by c³. Applying the quotient law first keeps the numerator together until its product is expanded.

Worked example 8. Given $y=a^4b^2/c^3$, with a, b and c positive, expand log y. All logarithms have the same fixed valid base.

Answer: $\log y=4\log a+2\log b-3\log c$. The complete working is:

1. Take logarithms of the given equality: $\log y=\log(a^4b^2/c^3)$.
2. Use the quotient law: $\log y=\log(a^4b^2)-\log(c^3)$.
3. Use the product law in the numerator: $\log y=\log(a^4)+\log(b^2)-\log(c^3)$.
4. Use the power law on each term: $\log y=4\log a+2\log b-3\log c$.

Taking logarithms of an equality means applying the same logarithm to both positive sides. The given positivity conditions ensure that y is positive. They also ensure that every individual logarithm appearing after expansion is defined.

How can the final expression be checked?

Match every original factor with one final term. Then check the exponent against the coefficient and the factor's position against the sign. This checks the structure without assigning numerical values to a, b or c.

Expansion need not produce a numerical answer. Here the letters are unspecified positive quantities. The expanded expression is the required algebraic answer. Assigning arbitrary values would replace the given general problem with a different numerical calculation.

You can expand the powers before separating the product, but each step must preserve the same expression. The quotient-first route makes the denominator's minus sign visible early, which helps prevent it from being lost.

How do you combine an expanded expression into one logarithm?

Combining logarithms, also called condensation, uses the same laws in reverse. A coefficient becomes an exponent, a sum of logarithms becomes a logarithm of a product, and a difference becomes a logarithm of a quotient. Each participating logarithm must have the same base.

Continue to use positive a, b and c, with a fixed valid logarithm base suppressed. Start from $4\log a+2\log b-3\log c$. The aim is to recover the single argument that has this logarithm.

Why should coefficients become powers first?

The product and quotient laws combine whole logarithms. First use the power law backwards so that each coefficient moves into the argument as an exponent. This makes the remaining addition and subtraction ready for the other laws.

Worked example 9. Combine $4\log a+2\log b-3\log c$ into one logarithm. Assume a, b and c positive and all logarithms to the same fixed valid base.

Answer: $\log(a^4b^2/c^3)$.

1. Move the coefficients into powers: $\log(a^4)+\log(b^2)-\log(c^3)$.
2. Combine the sum by the product law: $\log(a^4b^2)-\log(c^3)$.
3. Combine the difference by the quotient law: $\log(a^4b^2/c^3)$.

The answer is a logarithm of a quotient. It is not merely the quotient itself. Removing the logarithm would change the quantity being represented. If the original expression equals log y, then y is the positive argument inside that combined logarithm.

In the present expression, the terms added to one another supply numerator factors. The subtracted term supplies the denominator factor. Its coefficient becomes the power of that denominator factor; the subtraction sign does not make the factor negative.

How does reversing the calculation help?

Expand the combined result again. It must return to $4\log a+2\log b-3\log c$. This reverse check tests the powers, signs and factor placement together. It also shows that expansion and condensation are two directions of the same laws.

How can you choose the right method and check each answer?

Begin by reading the structure of the expression. A known equality between a power and a number suggests conversion. A product inside a logarithm suggests the product law. A quotient suggests subtraction, and a power suggests moving its exponent into a coefficient.

What should be checked before calculation?

Check the base and every argument. A valid logarithm base is positive and unequal to 1. Each real logarithm needs a positive argument. Before an expansion, check the smaller arguments that will appear afterwards as well as the original combined argument.

Next check that logarithms being combined share a base. The equalities involving 625 show why this matters: $\log_5 625=4$, whereas $\log_{25}625=2$. Identical arguments do not make logarithms with different bases interchangeable.

What should be checked after calculation?

Read conversions backwards. A logarithm value must be the exponent that reproduces its argument. For the equality involving 10000, the check is the given power 10⁴. Do not treat 10000 as the exponent simply because it stands beside the logarithm sign.

Reverse expansions. Move coefficients back to powers, combine added logarithms as products, and combine subtracted logarithms as quotients. The final argument should match the original expression. This is especially useful when the answer remains algebraic.

Finally, distinguish a logarithm from its argument. If the question asks for log y, an expression in logarithms can be the finished answer. If it asks for y and supplies log y, use the corresponding exponential relationship to recover y.

Note: The product, quotient and power laws do not give a corresponding rule for splitting the logarithm of a sum or difference. Identify the actual operation inside the brackets before choosing a law.

Glossary

• Logarithm — The exponent to which a specified valid base must be raised to produce a given positive argument.
• Base — The number raised to a power; a logarithm base must be positive and different from one.
• Argument — The quantity whose logarithm is being taken, required to be positive for a real logarithm.
• Exponent — The index specifying the power to which a base is raised in an exponential expression.
• Exponential form — A statement displaying a base raised to an exponent as equal to a resulting number.
• Logarithmic form — A statement expressing the exponent as the logarithm of the resulting positive number to its base.
• Common logarithm — A logarithm whose base is ten, with its value giving the required exponent of ten.
• Product law — The rule expressing a logarithm of positive factors multiplied together as the sum of their logarithms.
• Quotient law — The rule subtracting the denominator's logarithm from the numerator's logarithm when both arguments are positive.
• Power law — The rule moving an exponent of a positive logarithm argument outside as a coefficient.
• Expansion — Rewriting a combined logarithm as simpler logarithms joined by addition or subtraction with appropriate coefficients.
• Condensation — Combining logarithm terms with a common valid base into one logarithm by reversing the laws.

Common errors and misconceptions

• Misconception: The argument is the exponent in the converted exponential form. Correct: The logarithm's value is the exponent. In $\log_2 8=3$, the corresponding power is 2³ = 8.
• Misconception: A logarithm's base can be 1. Correct: Base 1 cannot uniquely identify an exponent, since every real power of 1 is 1. Use a positive base different from 1.
• Misconception: A negative logarithm value means the argument is invalid. Correct: The argument must be positive, but the logarithm itself is an exponent and may be negative.
• Misconception: A logarithm of a product is a product of logarithms. Correct: For positive arguments and the same base, the product law gives a sum of logarithms.
• Misconception: Division of arguments becomes division of logarithms. Correct: The quotient law gives subtraction, with the denominator's logarithm subtracted from the numerator's logarithm.
• Misconception: The power law raises the logarithm's value to the argument's exponent. Correct: It makes that exponent a coefficient multiplying the logarithm of the positive argument.
• Misconception: The product law also splits the logarithm of a sum. Correct: Its argument must be a product. Addition inside the argument does not justify adding separate logarithms.
• Misconception: The denominator term in the expansion of $a^4b^2/c^3$ is +3 log c. Correct: Division supplies subtraction, so the contribution is −3 log c.

Exam-style questions with model answers

Q1. Given 2³ = 8, write the equivalent logarithmic statement and identify its base, argument and value. [2 marks]

1. The equivalent logarithmic statement is $\log_2 8=3$, because the logarithm records the exponent in the given power relationship.
2. The base is 2, the argument is 8, and the logarithm's value is 3.

Q2. Convert $\log_{10}10000=4$ into exponential form and explain the roles of the three numbers. [2 marks]

1. The exponential form is 10⁴ = 10000, with the original logarithm base 10 unchanged.
2. The logarithm's value 4 becomes the exponent, while the argument 10000 is the result of raising 10 to that power.

Q3. Given 625 = 5⁴ = 25², find the logarithm of 625 to each of the bases 5 and 25. Explain why the answers differ. [3 marks]

1. Since the given equality includes 5⁴ = 625, the logarithmic definition gives $\log_5 625=4$. The value 4 records the exponent of 5.
2. Since it also includes 25² = 625, the same definition gives $\log_{25}625=2$. Here the exponent belongs to base 25.
3. The answers differ because the bases differ. The argument is 625 in both statements, but the required exponent depends on the chosen base.

Q4. Let p and q be positive real numbers and b a positive real number other than 1. Using the law of indices for multiplication, derive the product law of logarithms. [4 marks]

1. Let m and n be the logarithm values defined by $m=\log_b p$ and $n=\log_b q$, using the given common base.
2. Convert these definitions into exponential form: $p=b^m$ and $q=b^n$. Each positive argument is now expressed as a power.
3. Multiply the equalities and apply the index law: $pq=b^m b^n=b^{m+n}$. Multiplication of powers with the same base adds the exponents.
4. Convert back to logarithmic form: $\log_b(pq)=m+n=\log_b p+\log_b q$. This is the required product law under the stated conditions.

Q5. Let a, b and c be positive real numbers and let $y=a^4b^2/c^3$. Expand log y, naming the laws used and checking the conditions. All logarithms have the same fixed positive base other than 1. [5 marks]

1. The given positive values make a⁴, b² and c³ positive, so their quotient y is positive. Every logarithm needed in the expansion therefore has a positive argument.
2. Take logarithms of the equality and apply the quotient law: $\log y=\log(a^4b^2)-\log(c^3)$. The whole denominator contributes a subtracted logarithm.
3. Apply the product law to the numerator's logarithm: $\log y=\log(a^4)+\log(b^2)-\log(c^3)$. Both numerator factors contribute added logarithms.
4. Apply the power law to each term, moving its exponent outside: $\log y=4\log a+2\log b-3\log c$.
5. The coefficients retain the original powers 4, 2 and 3. The negative sign belongs to the c term because c³ was the denominator, completing the structural check.

Q6. Combine $4\log a+2\log b-3\log c$ into one logarithm, then state y if this expression equals log y. Assume a, b, c and y positive, with every logarithm to the same fixed positive base other than 1. [5 marks]

1. Use the power law in reverse to obtain $\log(a^4)+\log(b^2)-\log(c^3)$. Each coefficient becomes the exponent of its own positive argument.
2. Combine the two added logarithms by the product law, giving $\log(a^4b^2)-\log(c^3)$. The factors a⁴ and b² belong together in the numerator.
3. Apply the quotient law in reverse to obtain the single logarithm $\log(a^4b^2/c^3)$. The subtracted logarithm places c³ in the denominator.
4. Since this equals log y with the same valid base, the positive arguments agree: $y=a^4b^2/c^3$. This states y itself, rather than its logarithm.
5. Expanding the combined logarithm returns $4\log a+2\log b-3\log c$, confirming the original coefficients and signs. The positive assumptions ensure all separate logarithms used are defined.

Key takeaways

• A logarithm is an exponent: convert between exponential and logarithmic forms while preserving the base, exponent and positive argument.
• A real logarithm requires a positive argument and a positive base different from one; its value may be negative.
• The product law changes multiplication of positive arguments into addition of logarithms, provided the logarithms use the same valid base.
• The quotient law subtracts the denominator's logarithm from the numerator's logarithm; the order of subtraction matters.
• The power law moves an argument's exponent outside as a coefficient multiplying the logarithm of that positive argument.
• In the expansion of the given algebraic quotient, numerator powers become added terms and the denominator power supplies a subtracted term.
• Combine logarithms by reversing the laws: coefficients become powers, added terms supply products, and subtracted terms supply denominators.
• Check a converted answer using its exponential form, and check an expanded answer by combining its logarithms again.

Test yourself

What does $\log_b N=x$ mean, for a valid base b and positive argument N?

It means $b^x=N$: x is the exponent needed to obtain N from base b.

Given 2³ = 8, what is $\log_2 8$?

It is 3, because the logarithm records the exponent in the given equality 2³ = 8.

Why is base 1 excluded from real logarithms?

Every real power of 1 is 1, so a logarithm to that base cannot uniquely recover an exponent.

Why does $\log_b1=0$ for a valid base b?

The zero index gives $b^0=1$; converting that equality into logarithmic form gives the result.

For positive p and q and a valid base b, how is $\log_b(pq)$ expanded?

It becomes $\log_b p+\log_b q$, a sum of logarithms with the same base.

For positive p and q and a valid base b, which term is subtracted when expanding $\log_b(p/q)$?

The denominator's logarithm is subtracted, giving $\log_b p-\log_b q$ in that order.

For positive p and a valid base b, what does the power law give for $\log_b(p^2)$?

It gives $2\log_b p$; the exponent becomes a multiplier of the logarithm.

Given $y=a^4b^2/c^3$ with positive a, b and c, why is the log c term negative when all logarithms share a valid base?

The quotient law subtracts the denominator's logarithm, and the power law makes that contribution −3 log c.