Question

solve the equation. give an exact solution, and also approximate the solution to four decimal places. 5^{x - 3}=7 write the exact solution. x = (simplify your answer.)

Explanation:

Step1: Take the natural - logarithm of both sides

$\ln(5^{x - 3})=\ln(7)$

Step2: Use the power - rule of logarithms $\ln(a^b)=b\ln(a)$

$(x - 3)\ln(5)=\ln(7)$

Step3: Solve for $x$

$x-3=\frac{\ln(7)}{\ln(5)}$
$x = 3+\frac{\ln(7)}{\ln(5)}$

Answer:

$x = 3+\frac{\ln(7)}{\ln(5)}\approx3 + 1.2091=4.2091$
The exact solution is $x = 3+\frac{\ln(7)}{\ln(5)}$ and the approximate solution to four decimal places is $x\approx4.2091$