Question

solve. round your answer to the nearest thousandth.
$7 = 9^{x + 2}$
$x = \square$

Explanation:

Step1: Take logarithm on both sides

To solve for \( x \) in the equation \( 7 = 9^{x + 2} \), we can take the natural logarithm (ln) of both sides. This gives us \( \ln(7)=\ln(9^{x + 2}) \).

Step2: Use logarithm power rule

Using the power rule of logarithms, \( \ln(a^b)=b\ln(a) \), we can rewrite the right - hand side as \( (x + 2)\ln(9) \). So the equation becomes \( \ln(7)=(x + 2)\ln(9) \).

Step3: Solve for \( x+2 \)

We can solve for \( x + 2 \) by dividing both sides of the equation by \( \ln(9) \). So \( x+2=\frac{\ln(7)}{\ln(9)} \).

Step4: Solve for \( x \)

Now, we solve for \( x \) by subtracting 2 from both sides of the equation. \( x=\frac{\ln(7)}{\ln(9)}-2 \).

We know that \( \ln(7)\approx1.9459 \) and \( \ln(9)\approx2.1972 \). Then \( \frac{\ln(7)}{\ln(9)}\approx\frac{1.9459}{2.1972}\approx0.8857 \).

Subtracting 2 from this value: \( x\approx0.8857 - 2=- 1.1143\approx - 1.114 \) (rounded to the nearest thousandth).

Answer:

\( x\approx - 1.114 \)