Question

1/3log x-log5 as a single logarithm

Answers

GPT-4o mini
To combine the expression \(\frac{1}{3} \log x - \log 5\) into a single logarithm, we can follow these steps:

1. Rewrite \(\frac{1}{3} \log x\) using the property of logarithms that allows us to bring the coefficient in front into the logarithm as an exponent:
\[
\frac{1}{3} \log x = \log(x^{1/3}) = \log(\sqrt[3]{x})
\]

2. Now we can rewrite the expression:
\[
\log(\sqrt[3]{x}) - \log 5
\]

3. Use the property of logarithms that states \(\log a - \log b = \log\left(\frac{a}{b}\right)\):
\[
\log\left(\frac{\sqrt[3]{x}}{5}\right)
\]

Thus, the expression \(\frac{1}{3} \log x - \log 5\) as a single logarithm is:
\[
\log\left(\frac{\sqrt[3]{x}}{5}\right)
\]