To solve the equation [tex]\(2^{3x + 2} = 7^{x - 4}\)[/tex], follow these steps:
1. Express both sides in terms of logarithms:
Given:
[tex]\[
2^{3x + 2} = 7^{x - 4}
\][/tex]
Take the natural logarithm (ln) of both sides:
[tex]\[
\ln(2^{3x + 2}) = \ln(7^{x - 4})
\][/tex]
2. Apply properties of logarithms to simplify:
Using the property [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex], we get:
[tex]\[
(3x + 2) \cdot \ln(2) = (x - 4) \cdot \ln(7)
\][/tex]
3. Distribute the logarithms:
[tex]\[
3x \cdot \ln(2) + 2 \cdot \ln(2) = x \cdot \ln(7) - 4 \cdot \ln(7)
\][/tex]
4. Rearrange the equation to isolate x:
[tex]\[
3x \cdot \ln(2) - x \cdot \ln(7) = -4 \cdot \ln(7) - 2 \cdot \ln(2)
\][/tex]
5. Factor out x:
[tex]\[
x \cdot (3 \ln(2) - \ln(7)) = -4 \ln(7) - 2 \ln(2)
\][/tex]
6. Solve for x:
[tex]\[
x = \frac{-4 \ln(7) - 2 \ln(2)}{3 \ln(2) - \ln(7)}
\][/tex]
So, the solution set in terms of natural logarithms is:
[tex]\[
\left\{ \frac{-4 \ln(7) - 2 \ln(2)}{3 \ln(2) - \ln(7)} \right\}
\][/tex]
7. Calculate the numerical approximation:
Using a calculator:
[tex]\[
\ln(2) \approx 0.6931, \quad \ln(7) \approx 1.9459
\][/tex]
Substitute these values into the equation:
[tex]\[
x = \frac{-4 \cdot 1.9459 - 2 \cdot 0.6931}{3 \cdot 0.6931 - 1.9459}
\][/tex]
[tex]\[
x = \frac{-7.7836 - 1.3862}{2.0793 - 1.9459}
\][/tex]
[tex]\[
x = \frac{-9.1698}{0.1334}
\][/tex]
[tex]\[
x \approx -68.72
\][/tex]
Therefore, the decimal approximation for the solution is:
[tex]\[
\left\{ -68.72 \right\}
\][/tex]
In conclusion:
- The solution set expressed in terms of natural logarithms is [tex]\(\left\{ \frac{-4 \ln(7) - 2 \ln(2)}{3 \ln(2) - \ln(7)} \right\}\)[/tex].
- The solution set with a decimal approximation is [tex]\(\left\{ -68.72 \right\}\)[/tex].
