CHANGE OF BASE IN LOGARITHM

Before learning how to change of the base of a logarithm, you need to be aware of the multiplication of two logarithms.

Two logarithms can be multiplied, if one of the following two conditions is met.

Condition (i) :

The argument of the first logarithm and the base of the second logarithm must be same.

Condition (ii) :

The base of the first logarithm and the argument of the second logarithm must be same.

Example 1 :

Simplify : log_ab ⋅ log_bc

In the above two logarithms, the argument of the first logarithm and the base of the second logarithm are same. 

So, we can simplify the multiplication of above two logarithms by removing the part circled in red color. 

Example 2 :

Simplify : log_xy ⋅ log_zx

In the above two logarithms, the base of the first logarithm and the argument of the second logarithm are same.

So, we can simplify the multiplication of above two logarithms by removing the part marked in red color.

How to Change Base of a Logarithm

For some purposes, we may find it useful to change from logarithm in one base to logarithm in another base.

Suppose we are given log_ax and want to find log_bx.

We can write log_ax as a product of two logarithms with desired base in the first logarithm.

log_ax = log_bx ⋅ log_ab

Now, move the second logarithm (log_ab) in the above product to denominator. We already know, if a logarithm is moved from numerator to denominator or denominator to numerator, we have to switch the argument and base.

Then, we have

log_ax = (log_bx)/(log_ba)

Now, look at the base of the two logarithms on the right side, it is 'b'.

In this way, we can change the given base of a logarithm to any desired base.

But, always we don't have to write the given logarithm as a product of two logarithms. For understanding purpose, change of base is explained using the product of two logarithms above.

We can do this more quickly. Consider the logarithm given below.

log_ba

In the logarithm above, the base is 'b'. If we want to change the base, write two logarithms in division with the arguments 'a' in the logarithm in numerator and argument 'b' in the logarithm in denominator. And then, whatever base we want, we can take for the logarithms in numerator and denominator. But, the same base should be taken for both the logarithms.

For example, in log_ba, if we want to change the base to 'x', we can do it as shown below.

In the above example, the base of the given logarithm 'b' is changed to 'x'.

In this way, we can change the base of the given logarithm to any other base.

Solved Problems

Problem 1  :

If the value of log₁₀2 = 0.3010, find the value of log₂5. 

Solution :

Given : log₁₀2 = 0.3010.

The value of log₁₀2 is given and its base is 10.

So, change the base of log₂5 to 10.

log₂5 = (log₁₀5)/(log₁₀2)

= [log₁₀(10/2)]/(log₁₀2)

= (log₁₀10 - log₁₀2)/(log₁₀2)

= (1 - 0.3010)/0.3010

= 0.6990/0.3010

≈ 2.3223

Problem 2  :

Simplify the following expression and give your answer in terms of logarithm to the base 10.

Solution :

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