Questions: Solve the exponential equation and express the solution to the nearest hundredth. x=3^x=23

Solution Steps

To solve the exponential equation \(3^x = 23\), we can take the natural logarithm of both sides to linearize the equation. This allows us to solve for \(x\) using properties of logarithms.

1. Take the natural logarithm of both sides: \(\ln(3^x) = \ln(23)\).
2. Use the logarithm power rule: \(x \cdot \ln(3) = \ln(23)\).
3. Solve for \(x\): \(x = \frac{\ln(23)}{\ln(3)}\).
4. Use Python to compute the value of \(x\) to the nearest hundredth.

We start with the equation: \[ 3^x = 23 \] Taking the natural logarithm of both sides gives: \[ \ln(3^x) = \ln(23) \]

Using the power rule of logarithms, we can rewrite the left side: \[ x \cdot \ln(3) = \ln(23) \]

Now, we isolate \(x\) by dividing both sides by \(\ln(3)\): \[ x = \frac{\ln(23)}{\ln(3)} \] Calculating this gives: \[ x \approx 2.8540 \] Rounding to the nearest hundredth, we find: \[ x \approx 2.85 \]

Final Answer