To find the value of [tex]\( x \)[/tex] in the equation [tex]\( 3^{\frac{x}{4} + 2} = 56.8 \)[/tex], follow these steps:
1. Understand the Problem:
We need to solve for [tex]\( x \)[/tex] where the equation is [tex]\( 3^{\frac{x}{4} + 2} = 56.8 \)[/tex].
2. Isolate the Exponential Expression:
Rewrite the equation in a more convenient form:
[tex]\[
3^{\frac{x}{4} + 2} = 56.8
\][/tex]
3. Apply Logarithms:
To solve for [tex]\( x \)[/tex], take the natural logarithm (or log to any base) on both sides of the equation. Using the natural logarithm ([tex]\(\ln\)[/tex]), we get:
[tex]\[
\ln(3^{\frac{x}{4} + 2}) = \ln(56.8)
\][/tex]
4. Use Logarithm Properties:
Apply the power rule of logarithms, which states [tex]\(\ln(a^b) = b \ln(a)\)[/tex]:
[tex]\[
\left(\frac{x}{4} + 2\right) \ln(3) = \ln(56.8)
\][/tex]
5. Solve for [tex]\( \frac{x}{4} \)[/tex]:
Divide both sides by [tex]\(\ln(3)\)[/tex] to isolate the term [tex]\(\frac{x}{4} + 2\)[/tex]:
[tex]\[
\frac{x}{4} + 2 = \frac{\ln(56.8)}{\ln(3)}
\][/tex]
6. Isolate [tex]\( x \)[/tex]:
Subtract 2 from both sides of the equation:
[tex]\[
\frac{x}{4} = \frac{\ln(56.8)}{\ln(3)} - 2
\][/tex]
7. Solve for [tex]\( x \)[/tex]:
Multiply both sides of the equation by 4 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 4 \left( \frac{\ln(56.8)}{\ln(3)} - 2 \right)
\][/tex]
8. Approximate the Values:
Using a calculator to approximate the values, we find:
[tex]\[
\frac{\ln(56.8)}{\ln(3)} \approx 3.676944
\][/tex]
Subtract 2 from this result:
[tex]\[
3.676944 - 2 \approx 1.676944
\][/tex]
Multiply by 4:
[tex]\[
x \approx 4 \cdot 1.676944 \approx 6.707776
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is approximately:
[tex]\[
x \approx 6.708
\][/tex]
So the final answer is:
[tex]\[
x \approx 6.708
\][/tex]
