Question

Solve for x:
 
5 Superscript x Baseline equals one-third

Answer 1

To solve the equation \( 5^x = \frac{1}{3} \), we can use logarithms.

1. Take the logarithm of both sides. We can use either natural logarithm (ln) or common logarithm (log). Here, I will use the natural logarithm:

\[
\ln(5^x) = \ln\left(\frac{1}{3}\right)
\]

2. Apply the power rule of logarithms (\(\ln(a^b) = b \cdot \ln(a)\)):

\[
x \cdot \ln(5) = \ln\left(\frac{1}{3}\right)
\]

3. Solve for \(x\):

\[
x = \frac{\ln\left(\frac{1}{3}\right)}{\ln(5)}
\]

4. We can simplify \(\ln\left(\frac{1}{3}\right)\) using properties of logarithms \(\ln\left(\frac{1}{3}\right) = -\ln(3)\):

\[
x = \frac{-\ln(3)}{\ln(5)}
\]

Now, you can calculate \(x\) using a calculator:

- Approximately:
- \(\ln(3) \approx 1.0986\)
- \(\ln(5) \approx 1.6094\)

So,

\[
x \approx \frac{-1.0986}{1.6094} \approx -0.6826
\]

Therefore, the solution for \(x\) is:

\[
x \approx -0.6826
\]