Questions: For h(x)=3^x-5, what input value corresponds to h(x)=2? Round your answer to two decimal places.

Transcript text: For $h(x)=3^{x}-5$, what input value corresponds to $h(x)=2$ ? Round your answer to two decimal places.

Solution

Solution Steps

To find the input value $ x $ for which $ h(x) = 2 $, we need to solve the equation $ 3^x - 5 = 2 $. This simplifies to $ 3^x = 7 $. We can solve for $ x $ by taking the logarithm of both sides. Specifically, we use the natural logarithm to isolate $ x $.

Step 1: Set Up the Equation

We start with the function defined as $ h(x) = 3^x - 5 $. We need to find the value of $ x $ such that $ h(x) = 2 $. This leads us to the equation: \[ 3^x - 5 = 2 \]

Step 2: Simplify the Equation

Rearranging the equation gives: \[ 3^x = 7 \]

Step 3: Solve for $ x $

To solve for $ x $, we take the logarithm of both sides: \[ x = \frac{\log(7)}{\log(3)} \] Calculating this gives us approximately: \[ x \approx 1.7712 \] Rounding to two decimal places, we find: \[ x \approx 1.77 \]