Question

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

Explanation:

Step1: Take log of both sides

Take the logarithm (base 5 or natural log) of both sides. Let's use natural logarithm ($\ln$) for generality. So, $\ln(5^{5x - 4})=\ln(10)$.

Step2: Apply logarithm power rule

Using the power rule of logarithms, $\ln(a^b)=b\ln(a)$, we get $(5x - 4)\ln(5)=\ln(10)$.

Step3: Solve for $5x - 4$

Divide both sides by $\ln(5)$: $5x - 4=\frac{\ln(10)}{\ln(5)}$.

Step4: Solve for $5x$

Add 4 to both sides: $5x=\frac{\ln(10)}{\ln(5)} + 4$.

Step5: Solve for $x$

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 using logarithm properties: $\ln(10)=\ln(2\times5)=\ln(2)+\ln(5)$, so $x=\frac{\ln(2)+\ln(5)+4\ln(5)}{5\ln(5)}=\frac{\ln(2)+5\ln(5)}{5\ln(5)}=\frac{\ln(2)}{5\ln(5)} + 1$. Another way is to use change of base formula: $\frac{\ln(10)}{\ln(5)}=\log_5(10)$, so $x=\frac{\log_5(10)+4}{5}$.

Answer:

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