This lesson will show how to evaluate logarithms with some good examples. Study the first example below carefully.
Example #1:
Evaluate log_{8} 16
Write an equation in logarithmic form
log_{8} 16 = x
Convert the equation to exponential form
16 = 8^{x}
Write each side using base 2.
2^{4} = (2^{3})^{x}
2^{4} = 2^{3}^{x}
Since the base or 2 is the same, 2^{4} = 2^{3}^{x} if 4 is equal to 3x
If 4 = 3x, then x = 4/3
Therefore, log_{8} 16 = x = 4/3
Example #2:
Evaluate log_{10} 1000
Write an equation in logarithmic form
log_{10} 1000 = x
Convert the equation to exponential form
1000 = 10^{x}
Write each side using base 10.
10^{3} = (10)^{x}
Since the base or 10 is the same, 10^{3} = 10^{x} if 3 is equal to x
Therefore, log_{10} 1000 = x = 3
Example #3:
Evaluate log_{64} 1/16
Write an equation in logarithmic form
log_{64} 1/16 = x
Convert the equation to exponential form
1/16 = 64^{x}
Write each side using base 4.
1/4^{2} = (4^{3})^{x}
4^{-2} = 4^{3}^{x}
Since the base or 4 is the same, 4^{-2} = 4^{3}^{x} if -2 is equal to 3x
If -2 = 3x, then x = -2/3
Therefore, log_{64} 1/16 = x = -2/3
Example #4:
Evaluate log_{5} (-25)
Write an equation in logarithmic form
log_{5} (-25) = x
Convert the equation to exponential form
-25 = 5^{x}
No matter what x is, you could never get 5^{x} to equal to -25!
The logarithm of a negative number does not exist as a real number!
Example #5:
Evaluate log_{3} 10
Write an equation in logarithmic form
log_{3} 10 = x
Convert the equation to exponential form
3^{x} = 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
log_{10} 3^{x} = log_{10} 10
What is log_{10} 10 equal to?
In logarithmic form, log_{10} 10 is log_{10} 10 = y
In exponential form, log_{10} 10 = y is 10^{y} = 10 or 10^{y} = 10^{1}
So y = 1 and log_{10} 10 = 1
Substitute 1 for log_{10} 10 in log_{10} 3^{x} = log_{10} 10
We get log_{10} 3^{x} = 1
x log_{10} 3 = 1
x = 1/(log_{10} 3)
x = 1/0.477
x = 2.096
log_{3} 10 = 2.096
Sep 17, 23 09:46 AM
Jun 09, 23 12:04 PM