How to evaluate logarithms

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

How to evaluate logarithms

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

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