Question

solve the exponential equation. exp
$2(3)^{x-1}+1=57$

Explanation:

Step1: Isolate the exponential term

Subtract 1 from both sides.
$2(3)^{x-1} = 57 - 1$
$2(3)^{x-1} = 56$

Step2: Simplify the equation

Divide both sides by 2.
$(3)^{x-1} = \frac{56}{2}$
$(3)^{x-1} = 28$

Step3: Convert to logarithmic form

Use the logarithm base 3 on both sides.
$x - 1 = \log_{3}28$

Step4: Solve for x

Add 1 to both sides.
$x = \log_{3}28 + 1$

Step5: Optional decimal approximation

Use change of base formula $\log_{a}b=\frac{\ln b}{\ln a}$.
$x = \frac{\ln 28}{\ln 3} + 1 \approx \frac{3.3322}{1.0986} + 1 \approx 3.033 + 1 = 4.033$

Answer:

$x = \log_{3}28 + 1$ (or approximately $4.03$)