To solve the equation [tex]\(4^x = 20\)[/tex], we will use logarithms. Here is a step-by-step solution for this problem:
1. Start with the given equation:
[tex]\[
4^x = 20
\][/tex]
2. Apply the logarithm to both sides of the equation. It is often easiest to use the common logarithm (base 10) or the natural logarithm (base [tex]\(e\)[/tex]), but for this solution, we'll use the common logarithm (base 10):
[tex]\[
\log(4^x) = \log(20)
\][/tex]
3. Use the power rule of logarithms, which states that [tex]\(\log(a^b) = b \log(a)\)[/tex]:
[tex]\[
x \log(4) = \log(20)
\][/tex]
4. Isolate [tex]\(x\)[/tex] by dividing both sides of the equation by [tex]\(\log(4)\)[/tex]:
[tex]\[
x = \frac{\log(20)}{\log(4)}
\][/tex]
5. Now we need to evaluate the logarithms. Using a calculator:
[tex]\[
\log(20) \approx 1.3010
\][/tex]
[tex]\[
\log(4) \approx 0.6021
\][/tex]
6. Divide the two results to find [tex]\(x\)[/tex]:
[tex]\[
x = \frac{1.3010}{0.6021} \approx 2.1609640474436813
\][/tex]
7. Round the answer to two decimal places:
[tex]\[
x \approx 2.16
\][/tex]
Therefore, the solution to the equation [tex]\(4^x = 20\)[/tex] rounded to two decimal places is:
[tex]\[
\boxed{2.16}
\][/tex]
So, the correct answer is:
C. [tex]\( x = 2.16 \)[/tex]
