\( 3^x = 4^{x-1} \) then \(x = ?\)
Step-by-step Solution:
Given the equation \( 3^x = 4^{x-1} \), we solve for \( x \) as follows:
1. Take the logarithm of both sides:
\[
\log 3^x = \log 4^{x-1}
\]
2. Apply logarithm properties to simplify:
\[
x \log 3 = (x - 1) \log 4
\]
3. Expand and rearrange terms:
\[
x \log 3 = x \log 4 - \log 4
\]
\[
x (\log 4 - \log 3) = \log 4
\]
4. Solve for \( x \):
\[
x = \frac{\log 4}{\log 4 - \log 3}
\]
5. Simplify the expression:
\[
x = \frac{\log 2^2}{\log 2^2 - \log 3} = \frac{2 \log 2}{2 \log 2 - \log 3}
\]
6. Final expression for \( x \):
\[
x = \frac{2 \log 2}{2 \log 2 - \log 3}
\]
\[
x = \frac{2 \log 2}{2 \log 2 - 1}
\]