Algebra II: Exponents, Logarithms, \& Their Graphs

Question 11 of 20:

What is the solution to the equation below? Round your answer to two decimal places.

[tex] 4^x = 20 [/tex]

A. [tex] x = 2.99 [/tex]

B. [tex] x = 1.30 [/tex]

C. [tex] x = 2.16 [/tex]

D. [tex] x = 0.46 [/tex]



Answer :

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]