Question

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

Explanation:

Step1: Take logarithm on both sides

Take the natural - logarithm of both sides of the equation \(2^{x - 7}=3\). We get \(\ln(2^{x - 7})=\ln(3)\).

Step2: Use the power - rule of logarithms

According to the power - rule of logarithms \(\ln(a^b)=b\ln(a)\), so \((x - 7)\ln(2)=\ln(3)\).

Step3: Solve for x

First, divide both sides by \(\ln(2)\): \(x - 7=\frac{\ln(3)}{\ln(2)}\). Then add 7 to both sides: \(x = 7+\frac{\ln(3)}{\ln(2)}\).

Step4: Calculate the approximate value

We know that \(\ln(3)\approx1.0986\) and \(\ln(2)\approx0.6931\). Then \(\frac{\ln(3)}{\ln(2)}\approx\frac{1.0986}{0.6931}\approx1.5850\). So \(x\approx7 + 1.5850=8.5850\).

Answer:

Exact solution: \(x = 7+\frac{\ln(3)}{\ln(2)}\)
Approximate solution: \(x\approx8.5850\)