14) Solve for $x: 5^{x}=3^{x+1}$
a) $\frac{\log 3}{\log 5}$
b) $\bigcirc$
\[
\frac{\log 5-\log 3}{\log 3}
\]
c) 0
\[
\frac{\log 3}{\log 5-\log 3}
\]
d) 0
\[
\frac{\log 3}{\log 5+\log 3}
\]
Answer
\(\boxed{x \approx 2.15}\) is the final answer.
Step 1 ：Given the equation \(5^{x}=3^{x+1}\), we can take the logarithm of both sides to get \(x \log 5 = (x+1) \log 3\).
Step 2 ：Rearrange the equation to \(x \log 5 - x \log 3 = \log 3\).
Step 3 ：Factor out \(x\) on the left side of the equation to get \(x (\log 5 - \log 3) = \log 3\).
Step 4 ：Solve for \(x\) by dividing both sides of the equation by \((\log 5 - \log 3)\) to get \(x = \frac{\log 3}{\log 5 - \log 3}\).
Step 5 ：However, the actual solution is approximately \(x \approx 2.15\), which is not listed in the given options.
Step 6 ：\(\boxed{x \approx 2.15}\) is the final answer.
