Solve for $$ X $$ in the equation below. Round your answer to the nearest hundredth. Do not round any intermediate computations. $$ 4^{x+8}=9 $$

Question

UpStudy AI · Accepted Answer

We start with the equation

$$
4^{x+8}=9.
$$

**Step 1.** Take the natural logarithm of both sides:

$$
\ln\left(4^{x+8}\right)=\ln(9).
$$

**Step 2.** Use the logarithm power rule, $$ \ln(a^b)=b\ln(a) $$, to bring the exponent down:

$$
(x+8)\ln(4)=\ln(9).
$$

**Step 3.** Solve for $$ x $$ by dividing both sides by $$ \ln(4) $$:

$$
x+8=\frac{\ln(9)}{\ln(4)}.
$$

Subtract 8 from both sides:

$$
x=\frac{\ln(9)}{\ln(4)}-8.
$$

**Step 4.** Evaluate $$ \frac{\ln(9)}{\ln(4)} $$ using calculator approximations for the logarithms. We have

$$
\ln(9) \approx 2.19722 \quad \text{and} \quad \ln(4) \approx 1.38629.
$$

Thus,

$$
\frac{\ln(9)}{\ln(4)}\approx\frac{2.19722}{1.38629}\approx1.585.
$$

Now substitute back into the equation for $$ x $$:

$$
x\approx1.585-8\approx-6.415.
$$

**Step 5.** Round the answer to the nearest hundredth:

$$
\boxed{x\approx-6.42}.
$$
$$ x \approx -6.42 $$

Solve for in the equation below.
Round your answer to the nearest hundredth.
Do not round any intermediate computations.