Questions: Solve. Verify your result by checking that it satisfies or approximately satisfies the original equation. 3^(4x+7)=15 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. x= (Round to four decimal places as needed.) B. There is no real-number solution.

Solution Steps

To solve the equation \(3^{4x+7} = 15\), we can take the logarithm of both sides to bring down the exponent. This will allow us to solve for \(x\). After finding \(x\), we can verify the solution by substituting it back into the original equation to check if it approximately satisfies the equation.

We start with the equation: \[ 3^{4x + 7} = 15 \] Taking the logarithm of both sides gives us: \[ \log(3^{4x + 7}) = \log(15) \] Using the property of logarithms, we can rewrite the left side: \[ (4x + 7) \log(3) = \log(15) \]

Next, we isolate \(4x\): \[ 4x + 7 = \frac{\log(15)}{\log(3)} \] Subtracting 7 from both sides: \[ 4x = \frac{\log(15)}{\log(3)} - 7 \] Dividing by 4 gives us: \[ x = \frac{\frac{\log(15)}{\log(3)} - 7}{4} \]

Using the logarithmic values: \[ \log(3) \approx 1.0986 \quad \text{and} \quad \log(15) \approx 2.7081 \] We find: \[ x \approx \frac{2.7081 / 1.0986 - 7}{4} \approx -1.1338 \]

To verify, we substitute \(x\) back into the original equation: \[ 3^{4(-1.1338) + 7} \approx 3^{14.9999} \approx 15 \] This confirms that our solution is correct.

Final Answer