Question: 10. Solve $2^x = 7$

Solution

To solve the equation \(2^x = 7\), we’ll take the logarithm of both sides. Using the natural logarithm, the equation becomes: \[ \ln(2^x) = \ln(7) \] Apply the power rule of logarithms: \[ x \cdot \ln(2) = \ln(7) \] Now, solve for \(x\) by dividing both sides by \(\ln(2)\): \[ x = \frac{\ln(7)}{\ln(2)} \] Calculate the value using a calculator: \[ x \approx \frac{1.9459}{0.6931} \approx 2.807 \] So, \(x \approx 2.807\).