How to evaluate logarithms

This lesson will show how to evaluate logarithms with some good examples. Study the first example below carefully.

More examples showing how to evaluate logarithms


Example #1:

Evaluate log8 16

Write an equation in logarithmic form

log8 16 = x

Convert the equation to exponential form

16 = 8x

Write each side using base 2.

24 = (23)x

24 = 23x

Since the base or 2 is the same,   24 = 23x   if 4 is equal to 3x

If 4 = 3x, then  x = 4/3

Therefore, log8 16 = x = 4/3

Example #2:

Evaluate log10 1000

Write an equation in logarithmic form

log10 1000 = x

Convert the equation to exponential form

1000 = 10x

Write each side using base 10.

103 = (10)x

Since the base or 10 is the same,   103 = 10x   if 3 is equal to x

Therefore, log10 1000 = x = 3

Example #3:

Evaluate log64  1/16

Write an equation in logarithmic form

log64 1/16 = x

Convert the equation to exponential form

1/16 = 64x

Write each side using base 4.

1/42 = (43)x

4-2 = 43x

Since the base or 4 is the same,   4-2 = 43x   if -2 is equal to 3x

If -2 = 3x, then  x = -2/3

Therefore, log64  1/16 = x = -2/3

How to evaluate logarithms when it is not possible to write each side of the equation with the same base.


Example #4:

Evaluate log3  10

Write an equation in logarithmic form

log3  10 = x

Convert the equation to exponential form

3x = 10

Now, as you can see it is not possible to write each side with a base of 3 since it is very hard to rewrite 10 with a base of 3.

What we can do is to take the common logarithm of each side

log10 3x = log10 10

What is log10 10 equal to?

In logarithmic form, log10 10 is log10 10 = y

In exponential form, log10 10 = y is 10y = 10 or 10y = 101

So y  = 1 and log10 10 = 1

Substitute 1 for log10 10 in log10 3x = log10 10

We get log10 3x = 1

x log10 3 = 1

x = 1/(log10 3)

x = 1/0.477

x = 2.096

log3  10 = 2.096

Now study carefully the following figure with summarizes the concept

How to evaluate logarithms

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