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)
Examples
 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 nonstandardbase logarithm log _{a} (x) by converting it to a standardbase logarithm in fraction form log _{b} (x) / log _{b} (a) .
The nominator log _{b} (x) is the standardbase logarithm of the nonstandardbase logarithm’s exponent.
The denominator log _{b} (a) is the standardbase logarithm of nonstandardbase 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 Nonstandard Base Logarithm
Rewrite the nonstandard base logarithm as a fraction of a standard base logarithm
Frequently Asked Questions
lick 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?
Summary
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