日记 - logarithm change base

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日记 | ziyang-bai | logarithm change base

作者:

ziyang-bai (

2026/01/24 00:19)

阅读量: 9

Theorem
Prove that:
log_a(b)=log_c(b)/log_c(a)
where a, b, c > 0 and a, c unequal 1.
Answer:
Proof
Let x=log_a(b).
By the definition of a logarithm, we can rewrite this in exponential form:
a^x=b
Taking logarithms to the base c of both sides of the equation:
log_c(a^x)=log_c(b)
Using the power law of logarithms, loga(Mk)=klogaM:
xlog_c(a)=log_c(b)
Rearranging to make x the subject:
x=log_c(b)/log_c(a)
Hence, since x=logab, it follows that:
log_a(b)=log_c(b)/log_c(a)
■ Q.E.D

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8192Bit (2026/01/24 07:33)
Assume that frac{1}{log_b(a)}=log_a(b) and a^(log_a(b))=b are known conclusions.
frac{log_c(b)}{log_c(a)}
=log_c(b)*log_a(c)
According to the definition of logarithm
a^(log_c(b)*log_a(c))=b
(a^log_a(c))^log_c(b)=b
c^(log_c(b))=b
b=b
[INSERT STRANGE LATIN ABBREVIATION HERE]

日记 | ziyang-bai | logarithm change base