Question

Which expression represents 1/3 log x-log 5 as a single logarithm?

Answer 1

To combine the expression \( \frac{1}{3} \log x - \log 5 \) into a single logarithm, we can use the properties of logarithms.

1. **Use the power rule**: The term \( \frac{1}{3} \log x \) can be rewritten using the power property of logarithms, which states that \( a \log b = \log(b^a) \). Thus, we have:
\[
\frac{1}{3} \log x = \log(x^{1/3})
\]

2. **Rewrite the expression**:
The original expression can now be rewritten as:
\[
\log(x^{1/3}) - \log 5
\]

3. **Use the quotient rule**: According to the quotient property of logarithms, \( \log a - \log b = \log \left( \frac{a}{b} \right) \). Therefore:
\[
\log(x^{1/3}) - \log 5 = \log \left( \frac{x^{1/3}}{5} \right)
\]

Putting it all together, the expression \( \frac{1}{3} \log x - \log 5 \) as a single logarithm is:
\[
\log \left( \frac{x^{1/3}}{5} \right)
\]