How to evaluate logarithms
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This lesson will show how to evaluate logarithms with some good examples. Study the first example below carefully.
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
Example #4:
Evaluate log5 (-25)
Write an equation in logarithmic form
log5 (-25) = x
Convert the equation to exponential form
-25 = 5x
No matter what x is, you could never get 5x to equal to -25!
The logarithm of a negative number does not exist as a real number!
How to evaluate logarithms when it is not possible to write each side of the equation with the same base.
Example #5:
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
