Question

solve the following equation.
$5^{5x - 4} = 10$
$x = \square$ (type an exact answer.)

Explanation:

Step1: Take log on both sides

Take the logarithm (natural log or common log, here we use natural log) of both sides of the equation \(5^{5x - 4}=10\). We get \(\ln(5^{5x - 4})=\ln(10)\).
Using the logarithm power rule \(\ln(a^b)=b\ln(a)\), the left - hand side becomes \((5x - 4)\ln(5)=\ln(10)\).

Step2: Solve for x

First, divide both sides of the equation \((5x - 4)\ln(5)=\ln(10)\) by \(\ln(5)\):
\(5x-4 = \frac{\ln(10)}{\ln(5)}\)
Then, add 4 to both sides:
\(5x=\frac{\ln(10)}{\ln(5)} + 4\)
Finally, divide both sides by 5:
\(x=\frac{\frac{\ln(10)}{\ln(5)}+4}{5}=\frac{\ln(10)+4\ln(5)}{5\ln(5)}\)
We can also simplify \(\frac{\ln(10)+4\ln(5)}{5\ln(5)}\) using the property \(a\ln(b)=\ln(b^a)\) and \(\ln(a)+\ln(b)=\ln(ab)\).
Since \(4\ln(5)=\ln(5^4)=\ln(625)\) and \(\ln(10)+\ln(625)=\ln(10\times625)=\ln(6250)\), so \(x = \frac{\ln(6250)}{5\ln(5)}\) or \(x=\frac{\log_5(10)+4}{5}\) (using the change - of - base formula \(\frac{\ln(a)}{\ln(b)}=\log_b(a)\)).

Answer:

\(\frac{\log_{5}(10)+4}{5}\) (or equivalent forms like \(\frac{\ln(10)+4\ln(5)}{5\ln(5)}\) or \(\frac{\ln(6250)}{5\ln(5)}\))