Solve the exponential equation. Express irrational soluti $$ 5^{x-4}=3^{2 x} $$ A. The solution set is $$ \} $$. (Round to the nearest thousandth as needed. Us B. The solution is the empty set.

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To solve the exponential equation $$ 5^{x-4} = 3^{2x} $$, we can take the logarithm of both sides. We will use the natural logarithm (ln), but you can use any logarithm.

Starting with the equation:

$$
5^{x-4} = 3^{2x}
$$

Taking the natural logarithm of both sides:

$$
\ln(5^{x-4}) = \ln(3^{2x})
$$

Using the property of logarithms that allows us to bring the exponent down:

$$
(x-4) \ln(5) = 2x \ln(3)
$$

Now, distribute the logarithm on the left side:

$$
x \ln(5) - 4 \ln(5) = 2x \ln(3)
$$

Next, we can rearrange the equation to isolate $$x$$:

$$
x \ln(5) - 2x \ln(3) = 4 \ln(5)
$$

Factor out $$x$$ on the left side:

$$
x (\ln(5) - 2 \ln(3)) = 4 \ln(5)
$$

Now, solve for $$x$$:

$$
x = \frac{4 \ln(5)}{\ln(5) - 2 \ln(3)}
$$

Now we can calculate the value of $$x$$ using a calculator. First, we need to find the values of $$\ln(5)$$ and $$\ln(3)$$:

$$
\ln(5) \approx 1.6094
$$
$$
\ln(3) \approx 1.0986
$$

Now substitute these values into the equation:

$$
x = \frac{4 \cdot 1.6094}{1.6094 - 2 \cdot 1.0986}
$$

Calculating the denominator:

$$
1.6094 - 2 \cdot 1.0986 = 1.6094 - 2.1972 = -0.5878
$$

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

$$
x = \frac{4 \cdot 1.6094}{-0.5878} \approx \frac{6.4376}{-0.5878} \approx -10.95
$$

Rounding to the nearest thousandth, we get:

$$
x \approx -10.950
$$

Thus, the solution set is:

$$
\{ -10.950 \}
$$
The solution is $$ x \approx -10.950 $$.

Solve the exponential equation. Express irrational soluti \( 5^{x-4}=3^{2 x} \) A. The solution set is \( \} \). (Round to the nearest thousandth as needed. Us B. The solution is the empty set.

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