Table of Contents
The change of base rule for logarithms is:
log a (x) = log b (x) / log b (a) a,b,x ∈ R + a,b ≠ 1
R + denotes the set of positive real numbers.
It is important to understand that b can be any valid base including, but not limited, to 2,10 and e.
Change of Base Rule Proof
b x = a ⇒ x = log b(a)
We take the logarithm base d on both sides:
log d(b) x = log d(a)
We apply the logarithm power rule:
log d(b) x = x log d(b)
x log d(b) = log d(a) ⇒ x = log d(a) / log d(b)
Because of our assumption:
x = log b(a) ⇒ log b(a) = log d(a) / log d(b)
- b = 2 Log a (x) = log 2 (x) / log 2 (a) Log 4 (16) = log 2 (16) / log 2 (4) = 4 / 2 = 2
- b = 10 Log a (x) = log (x) / log (a) Log 4 (16) = log (16) / log (4) = 1.2041199826 / 0.6020599913 = 2
- b = e Log a (x) = ln (x) / ln (a) Log 4 (16) = ln(16) / ln (4) = 2.7725887222 / 1.3862943611 = 2
- b = 5 Log a (x) = log 5 (x) / log 5 (a) Log 4 (16) = log 5 (16) / log 5 (4) = 1.7227062322 / 0.8613531161 = 2
- b = 8 Log a (x) = log 8 (x) / log 8 (a) Log 4 (16) = log 8 (16) / log 8 (4) = 1.3333333333 / 0.6666666666 = 2
Change of Base Rule in Logarithms
Our examples demonstrate that you can evaluate a non-standard-base logarithm log a (x) by converting it to a standard-base logarithm in fraction form log b (x) / log b (a) .
The nominator log b (x) is the standard-base logarithm of the non-standard-base logarithm’s exponent.
The denominator log b (a) is the standard-base logarithm of non-standard-base logarithm’s base.
How to Change the Base of a Log
Decide on the Standard Base
Decide which standard base you are going to use
Rewrite the Non-standard Base Logarithm
Rewrite the non-standard base logarithm as a fraction of a standard base logarithm
Frequently Asked Questionslick on the question which is of interest to you to see the collapsible content answer.
How do you use the change of base formula?
What is the change of base process?
How to find the base of a logarithmic function?
What is the change of base property?
Can you change the base of a log?
How do you change the base of a log on a calculator?
What does change of base formula create?
How does the change of base rule work?
If you have not already done so, please hit the share buttons, and install our PWA app (see sidebar or menu).
If you still do things the traditional way: bookmark us now! BTW: Here’s the Logarithm Power Rule.
Thanks for your visit!