Solve the given equation for $$ x $$ $$ 7^{4 x-5}=32 $$ $$ x=\square $$

Question

UpStudy AI · Accepted Answer

We start with the equation

$$
7^{4x-5} = 32.
$$

**Step 1. Apply the logarithm to both sides.**

Taking the natural logarithm of both sides, we have

$$
\ln\left(7^{4x-5}\right) = \ln(32).
$$

**Step 2. Use the power rule of logarithms.**

The logarithm power rule gives

$$
(4x-5)\ln(7)=\ln(32).
$$

**Step 3. Solve for $$ 4x-5 $$.**

Divide both sides by $$\ln(7)$$:

$$
4x-5=\frac{\ln(32)}{\ln(7)}.
$$

**Step 4. Solve for $$ x $$.**

Add 5 to both sides:

$$
4x=\frac{\ln(32)}{\ln(7)}+5.
$$

Then, divide by 4:

$$
x=\frac{1}{4}\left(\frac{\ln(32)}{\ln(7)}+5\right).
$$

Thus, the solution is

$$
x=\frac{5}{4}+\frac{\ln(32)}{4\ln(7)}.
$$
$$ x = \frac{5}{4} + \frac{\ln(32)}{4\ln(7)} $$

Solve the given equation for