Table of Contents
The change of base rule for logarithms is:
loga (x) = logb (x) / logb (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
bx = a ⇒ x = logb(a)
We take the logarithm base d on both sides:
logd(b)x = logd(a)
We apply the logarithm power rule:
logd(b)x = x logd(b)
x logd(b) = logd(a) ⇒ x = logd(a) / logd(b)
Because of our assumption:
x = logb(a)⇒ logb(a) = logd(a) / logd(b)
- b = 2 Loga (x) = log2 (x) / log2 (a) Log4 (16) = log2 (16) / log2 (4) = 4 / 2 = 2
- b = 10 Loga (x) = log (x) / log (a) Log4 (16) = log (16) / log (4) = 1.2041199826 / 0.6020599913 = 2
- b = e Loga (x) = ln (x) / ln (a) Log4 (16) = ln(16) / ln (4) = 2.7725887222 / 1.3862943611 = 2
- b = 5 Loga (x) = log5 (x) / log5 (a) Log4 (16) = log5 (16) / log5 (4) = 1.7227062322 / 0.8613531161 = 2
- b = 8 Loga (x) = log8 (x) / log8 (a) Log4 (16) = log8 (16) / log8 (4) = 1.3333333333 / 0.6666666666 = 2
Change of Base Rule in Logarithms
Our examples demonstrate that you can evaluate a non-standard-base logarithm loga (x) by converting it to a standard-base logarithm in fraction form logb (x) / logb (a).
The nominator logb (x) is the standard-base logarithm of the non-standard-base logarithm’s exponent.
The denominator logb (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.
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What is the change of base property?
Can you change the base of a log?
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How does the change of base rule work?
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