Question: Solve the following elementary exponential equation. \[ 32^x - 64 = 0 \] Enter your answer…

Solve the following elementary exponential equation.

\[ 32^x - 64 = 0 \]

Enter your answer (opens in new window)

\[ x = \boxed{\phantom{0}} \]

Solution

To solve the equation \(3^{2x} - 64 = 0\), follow these steps: First, isolate the exponential expression: \[ 3^{2x} = 64 \] Take the logarithm of both sides to solve for \(x\). Using the natural logarithm: \[ \ln(3^{2x}) = \ln(64) \] Apply the logarithmic identity \(\ln(a^b) = b\ln(a)\): \[ 2x \cdot \ln(3) = \ln(64) \] Solve for \(x\) by dividing both sides by \(2 \ln(3)\): \[ x = \frac{\ln(64)}{2 \ln(3)} \] This is the solution for \(x\).