Certainly! Let's solve the equation [tex]\(5^{x+1} = 8\)[/tex].
1. Rewrite the Equation:
Start with the given equation:
[tex]\[
5^{x+1} = 8
\][/tex]
2. Apply the Logarithm:
To isolate [tex]\(x\)[/tex], take the natural logarithm (or logarithm to any base, but we'll use natural logarithm for simplicity) of both sides:
[tex]\[
\ln(5^{x+1}) = \ln(8)
\][/tex]
3. Use the Logarithm Power Rule:
Apply the logarithm power rule, [tex]\(\ln(a^b) = b \ln(a)\)[/tex], to the left-hand side:
[tex]\[
(x + 1) \ln(5) = \ln(8)
\][/tex]
4. Solve for x:
Isolate [tex]\(x\)[/tex] by first removing the logarithm term involving [tex]\(x\)[/tex]:
[tex]\[
x + 1 = \frac{\ln(8)}{\ln(5)}
\][/tex]
5. Final Step:
Solve for [tex]\(x\)[/tex] by subtracting 1 from both sides:
[tex]\[
x = \frac{\ln(8)}{\ln(5)} - 1
\][/tex]
The solution to the equation [tex]\(5^{x+1} = 8\)[/tex] is:
[tex]\[
x = \frac{\ln(8)}{\ln(5)} - 1
\][/tex]
To simplify further, we can express the logarithm ratio explicitly:
[tex]\[
x = \log_5\left(\frac{8}{5}\right)
\][/tex]
Therefore, the solution is:
[tex]\[
x = \log \left( \left(\frac{8}{5}\right)^{1/\ln(5)} \right)
\][/tex]
This form matches the provided outcome, confirming our solution.
