Express one logarithm as a ratio in a common base.
Notes and key methods
Writing \(\displaystyle u=\log _{a}b\) gives \(\displaystyle a^{u}=b\). Taking logarithms in any valid common base c yields \(u\log_c a=\log_c b\), so \(\displaystyle \log_{a} b= \frac{\log_{c} b}{\log_{c} a}\).
Both a and c must be positive and different from one, and b must be positive. Reciprocal log relationships follow only for such valid bases and arguments.
Check the conditions
Both logarithm bases are positive and different from one.
The argument is positive.
Three worked examples
Example 1 · Changing the logarithm base
Let \(\displaystyle a=16\) and \(\displaystyle b=32\). Use base-2 logarithms to work exactly.
Find \(\displaystyle \log _{a}b\).
Find log_b a.
Find log_(\(\displaystyle a^{2}\)) b.
Find log_b(\(\displaystyle a^{2}\)).
Show the reasoning
(a)
Take base-2 logarithms of the argument and divide by that of the base.
\(\displaystyle \frac{5}{4}\)
Answer: \(\displaystyle \frac{5}{4}\).
(b)
Take base-2 logarithms of the argument and divide by that of the base.
\(\displaystyle \frac{4}{5}\)
Answer: \(\displaystyle \frac{4}{5}\).
(c)
Take base-2 logarithms of the argument and divide by that of the base.
\(\displaystyle \frac{5}{8}\)
Answer: \(\displaystyle \frac{5}{8}\).
(d)
Take base-2 logarithms of the argument and divide by that of the base.
\(\displaystyle \frac{8}{5}\)
Answer: \(\displaystyle \frac{8}{5}\).
Example 2 · Changing the logarithm base
Let \(\displaystyle a=32\) and \(\displaystyle b=64\). Use base-2 logarithms to work exactly.
Find \(\displaystyle \log _{a}b\).
Find log_b a.
Find log_(\(\displaystyle a^{2}\)) b.
Find log_b(\(\displaystyle a^{2}\)).
Show the reasoning
(a)
Take base-2 logarithms of the argument and divide by that of the base.
\(\displaystyle \frac{6}{5}\)
Answer: \(\displaystyle \frac{6}{5}\).
(b)
Take base-2 logarithms of the argument and divide by that of the base.
\(\displaystyle \frac{5}{6}\)
Answer: \(\displaystyle \frac{5}{6}\).
(c)
Take base-2 logarithms of the argument and divide by that of the base.
\(\displaystyle \frac{3}{5}\)
Answer: \(\displaystyle \frac{3}{5}\).
(d)
Take base-2 logarithms of the argument and divide by that of the base.
\(\displaystyle \frac{5}{3}\)
Answer: \(\displaystyle \frac{5}{3}\).
Example 3 · Changing the logarithm base
Let \(\displaystyle a=64\) and \(\displaystyle b=8\). Use base-2 logarithms to work exactly.
Find \(\displaystyle \log _{a}b\).
Find log_b a.
Find log_(\(\displaystyle a^{2}\)) b.
Find log_b(\(\displaystyle a^{2}\)).
Show the reasoning
(a)
Take base-2 logarithms of the argument and divide by that of the base.
\(\displaystyle \frac{1}{2}\)
Answer: \(\displaystyle \frac{1}{2}\).
(b)
Take base-2 logarithms of the argument and divide by that of the base.
\(2\)
Answer: \(2\).
(c)
Take base-2 logarithms of the argument and divide by that of the base.
\(\displaystyle \frac{1}{4}\)
Answer: \(\displaystyle \frac{1}{4}\).
(d)
Take base-2 logarithms of the argument and divide by that of the base.
\(4\)
Answer: \(4\).
Common mistakes
The denominator is the logarithm of the original base.
Do not invert only one part of a change-of-base expression.