Change of Base Formula: Derivation and Solved Examples

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The change of base formula is used to change the base of a logarithm.

A logarithm is defined as the power to which a base must be raised to yield a given number.
We may have noticed two buttons in scientific calculators: "log" and "ln".
"log" is a base 10 logarithm, while "ln" is a base e logarithm.
However, calculating the logarithm of a number using bases other than 10 and e is not an option.
This problem is solved by changing the base formula.
It is also used for solving many logarithmic problems.

Table of Content

What is Change of Base Formula?
Change of Base Formula Derivation
Solved Examples of Using Change of Base Formula
Things to Remember
Sample Questions

Key Terms: Logarithm, Change of base formula, Ratio, Power, ln, log, Logarithm formula, Exponent, Exponential, Base, Number, Numerator, Denominator.

What is Change of Base Formula?

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The change of base formula is used to express a logarithm of a number with a specific base as the ratio of two logarithms, each with a base that differs from the original logarithm. This is a property of logarithms.

Change of Base Formula

The basic logarithm with a base is converted into two logarithms with a new and similar base. The change of base formula is given by

\(\LARGE \log_{a}b=\frac{\log _{c}b}{\log _{c}a}\)

The above formula can also be written as

\(\LARGE \log_{a}b \: .\: \log _{c}a = \log _{c}b\)

In the above formula

The argument of the logarithm in the numerator is the same as that of the original logarithm.
The argument of the logarithm in the denominator is the same as that of the base of the original logarithm.
Both the numerator and denominator logarithms should have the same base, which can be any positive integer other than 1.

Change of Base Formula

Change of Base Formula

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Change of Base Formula Derivation

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The following steps can help to understand the change in base formula derivation.

Let us assume that,

\(\log_{b}a = p\)
\(\log_{c}a = q\)
\(\log_{c}b = r\)

On converting the above expressions into exponential form, we get

\(a =b^{\:p}, \: a=c^{\:q}, \:and \: b=c^ {\:q}\)

From the above first two equations, we have

\(b^{\:p} = c^{\:q}\)

Substituting \(b= c^{\:r}\), we get

\((c^{\:r})^p = c^{\:q}\)

\(\Rightarrow c^{\:rp} = c^{\:q}\)

\(\Rightarrow rp = q\)

\(\Rightarrow p= \frac {q}{r}\)

Substituting the values of p, q, and r, we get

\(\log_{b}a = \frac {\log_{c}a}{\log_{c}b}\)

Change of base formula pdf notes:

Solved Examples of Using Change of Base Formula

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Ques. Solve \(3\log_{4}30\).

Ans. Given \(3\log_{4}30\)

Using the formula, \( \log_{b}a=\frac{\log _{c}a}{\log _{c}b}\), we get

\(3\log_{4}30 = 3 \times \frac {\log_{10}30 }{\log_{10}4 }\)

\(\Rightarrow 3\log_{4}30 = 3 \times \frac {1.48}{0.60}\)

\(\Rightarrow 3\log_{4}30 = 7.4\)

Ques. Solve \(\log_{32}16\).

Ans. Given \(\log_{32}16\)

Using the formula, \( \log_{b}a=\frac{\log _{c}a}{\log _{c}b}\), we get

\(\log_{32}16 = \frac {\log_{10}16 }{\log_{10}32 }\)

\(\Rightarrow \log_{32}16 = \frac {1.2}{1.5}\)

\(\Rightarrow \log_{32}16 = 0.8\)

Things to Remember

A logarithm is defined as the power to which a base must be raised to yield a given number.
"log" is a base 10 logarithm, while "ln" is a base e logarithm.
The basic logarithm with a base is converted into two logarithms with a new and similar base.
The change of base formula is given by \(\log_{b}a=\frac{\log _{c}a}{\log _{c}b}\).
Both the numerator and denominator logarithms should have the same base, which can be any positive integer other than 1.

Also Read:

Related Articles
Log Base 2	Logarithms	Logarithms Formula
Value of Log 1 to 10	Logarithmic Functions	Value of Log 1

Sample Questions

Ques. What are the applications of the change of base formula in logarithms? (2 Marks)

Ans. The change of base formula is mostly used to convert the base of a logarithm to any other base. This is commonly used to calculate logarithms in bases other than 10 and "e" because the calculator only includes choices for calculating logarithms in bases 10 (log button) and e (ln button).

Ques. How to use Change of Base Formula? (2 Marks)

Ans. The basics of the formula change to the base is log b = [log_c a] / [log_c b]. To change the base of a logarithm, divide [log a] by [log b]. These logarithms can have any positive integer as their base.

Ques. Solve the following logarithm: 2log₄ 29. (2 Marks)

Ans. The given logarithm is, 2log₄ 29

Now using the change of base formula,

log _b a = [log _c a] / [log _c b]

2 log₄ 29 = 2 × log₁₀ 29 / log ₁₀ 4

2log₄ 29 = 2 x 2.43 = 4.86

Ques. Simplify the given logarithm: log₃₂ 16. (2 Marks)

Ans. The logarithm given is,

log ₃₂ 16

Thus using a change of base formula,

log _b a = [log _c a] / [log _c b]

log ₃₂ 16 = log ₁₀ 16 / log ₁₀ 32

= log ₁₀ 24 / log ₁₀ 25

= 4 log ₁₀ 2 / 5 log ₁₀ 2 [since, we know, log a^m = m log a]

= 4 / 5

Ques. Evaluate the value of log ₆₄ 8 using the change of base formula. (3 Marks)

Ans. Here the change of base formula is used to change the base to 10.

log _b a = [log _c a] / [log _c b]

We know that log 10 x is the same as log x.

log ₆₄ 8 = log ₁₀ 8 / log ₁₀ 64

= log 8 / log 82

= log 8 / 2 log 8 [since, we know, log a^m = m log a]

= 1 / 2

Ques. Calculate log ₉ 8. Round your answer to 4 decimals. (2 Marks)

Ans. We cannot calculate log 9 8 directly so we apply the change of base formula to change the base to 10 so it can be calculated easily.

log _b a = [log _c a] / [log _c b]

log ₉ 8 = log ₁₀ 8 / log ₁₀ 9

= 0.903 / 0.954

= 0.9465

Ques. Calculate the value of log 6 10 using the change of base formula. (2 Marks)

Ans. From the change of base formula, we have

log _b a = [log _c a] / [log _c b]

log ₆ 10 = log ₁₀ 10 / log ₁₀ 6

= 1 / 0.778

= 1.285

Ques. Calculate the value of log ₄ 10 using the change of base formula. (2 Marks)

Ans. From the change of base formula, we have

log _b a = [log _c a] / [log _c b]

log ₄ 10 = log ₁₀ 10 / log ₁₀ 4

= 1 / 0.602

= 1.66

Ques. Calculate the value of log ₃ 2 · log ₄ 3 · log ₅ 4 using necessary logarithm functions and formulae. (5 Marks)

Ans. Here we have to use the change of base formula in the multiplication form which is,

log _b a · log _c b = log _c a

We apply this formula twice to evaluate the given logarithmic expression.\

The given logarithm expression is,

log ₃ 2 · log ₄ 3 · log ₅ 4

Solving the first two terms, we get,

log ₃ 2 · log ₄ 3 · log ₅ 4

Here, a=2, b=3, c=4;

Therefore, according to, log _b a · log _c b = log _c a, we get,

log ₃ 2 · log ₄ 3 = log ₄ 2

Substituting this value in the given expression, we have

= log ₄ 2 · log ₅ 4

Now, on solving these two terms, we get,

log ₄ 2 · log ₅ 4

Here, a = 2, b = 4, c = 5;

Therefore, according to, log b a · log c b = log c a, we get,

log ₄ 2 · log ₅ 4 = log ₅ 2

Therefore, the value of log ₃ 2 · log ₄ 3 · log ₅ 4 is log ₅ 2.

Now we use the change of base in this form to change the base of this log 5 2 to 10 for further calculation.

log _b a = [log _c a] / [log _c b]

Here, a = 2, b = 5, c = 10.

Thus, log ₅ 2 = log ₁₀ 2 / log ₁₀ 5

= 0.301 / 0.69897

= 0.99997

Ques. Evaluate the given logarithm and round your answer to the nearest thousandth. log ₂ (1/50). (3 Marks)

Ans. The logarithm given is,

log ₂ (1/50)

Thus by using the change of base formula,

log _b a = [log _c a] / [log _cb]

log ₂ (1/50) = log ₁₀ (1/50) / log ₁₀ 2

= (log ₁₀ 1 - log 10 50) / log ₁₀ 2 [since we know log (a/b) = log a - log b]

= - log ₁₀ 50 / log ₁₀ 2 [since we know log ₁₀ 1= 0]

= - 5.644

Ques. Evaluate the given logarithm and round your answer to the nearest thousandth. log ₃ (0.2). (3 Marks)

Ans. The logarithm given is,

log ₃ (0.2)

Thus using the change of base formula,

log _b a = [log _c a] / [log _c b]

log ₃ (0.2)

= log ₁₀ (0.2) / log ₁₀ 3

Now, log ₁₀ (0.2) = log ₁₀ (1/5)

= (log ₁₀ 1 - log ₁₀ 5) / log ₁₀ 2 [since we know log (a/b) = log a - log b]

= - log ₁₀5 / log ₁₀ 2 [since we know log ₁₀ 1= 0]

= - 1.465

Ques. Evaluate the given logarithm and round your answer to the nearest thousandth. log ₇ (25). (3 Marks)

Ans. The logarithm given is,

log ₇ 25

Thus using the change of base formula,

log _b a = [log _c a] / [log _c b]

log ₇ 25

= log ₁₀ 25 / log ₁₀ 7

Now, log ₁₀ 25 = log ₁₀ 52

= log ₁₀ 52 / log ₁₀ 2

= 2log ₁₀ 5 / log ₁₀ 2 [since, we know, log a^m = m log a]

= 1.654

Ques. Evaluate the value of log ₃ 2 · log ₄ 3 · log ₅ 4. (2 Marks)

Ans. Using the change of base formula, log _b a ⋅ log _c b = log _c a.

We get

log ₃ 2 · log ₄ 3 · log ₅ 4

= log ₄ 2 · log ₅ 4

= log ₅ 2

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Change of Base Formula: Derivation and Solved Examples

What is Change of Base Formula?

Change of Base Formula

Change of Base Formula Derivation

Solved Examples of Using Change of Base Formula

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

2.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

3.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

4.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

5.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

Comments

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Change of Base Formula: Derivation and Solved Examples

What is Change of Base Formula?

Change of Base Formula

Change of Base Formula Derivation

Solved Examples of Using Change of Base Formula

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1. Find: If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

2.Find:Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

3.If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

4.Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

5. Find: The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

2.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

3.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

4.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

5.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is: