Let's solve the equation [tex]\(2^{\frac{x}{3}} = 100.6\)[/tex] for [tex]\(x\)[/tex] step by step.
1. Understand the Equation:
We start with the equation:
[tex]\[
2^{\frac{x}{3}} = 100.6
\][/tex]
2. Apply Logarithms:
To get [tex]\(x\)[/tex] out of the exponent, we take the natural logarithm (or logarithm to any consistent base) on both sides:
[tex]\[
\log\left(2^{\frac{x}{3}}\right) = \log(100.6)
\][/tex]
3. Use Logarithm Properties:
Using the property of logarithms that [tex]\(\log(a^b) = b \cdot \log(a)\)[/tex], we can rewrite the left-hand side:
[tex]\[
\frac{x}{3} \log(2) = \log(100.6)
\][/tex]
4. Isolate [tex]\(x\)[/tex]:
To solve for [tex]\(x\)[/tex], we first isolate [tex]\(\frac{x}{3}\)[/tex]:
[tex]\[
\frac{x}{3} = \frac{\log(100.6)}{\log(2)}
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Finally, multiply both sides by 3 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 3 \cdot \frac{\log(100.6)}{\log(2)}
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] can be computed directly using logarithms.
6. Numerical Solution:
Plugging the values into a calculator, we find:
[tex]\[
x \approx 19.96
\][/tex]
So, the solution to the equation [tex]\(2^{\frac{x}{3}} = 100.6\)[/tex] is approximately:
[tex]\[
x \approx 19.96
\][/tex]
