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

\[ 3 \cdot e^x = 11.76 \]

A. \[ x = 0.59 \]

B. \[ x = 50.40 \]

C. \[ x = 1.37 \]

D. \[ x = 40.98 \]

Answer :

To solve the equation [tex]\(3 \cdot e^x = 11.76\)[/tex] for [tex]\(x\)[/tex], follow these steps:

1. Isolate the exponential expression: Divide both sides of the equation by 3 to get the exponential expression by itself.

[tex]\[
e^x = \frac{11.76}{3}
\][/tex]

2. Simplify the division: Calculate the result of the division.

[tex]\[
e^x = 3.92
\][/tex]

3. Use natural logarithms to solve for [tex]\(x\)[/tex]: Take the natural logarithm (ln) of both sides. The natural logarithm is the inverse operation of the exponential function, so this step will help us solve for [tex]\(x\)[/tex].

[tex]\[
x = \ln(3.92)
\][/tex]

4. Compute the natural logarithm: Find the natural logarithm of 3.92. When computed, this gives a value approximately equal to:

[tex]\[
x \approx 1.366091653802371
\][/tex]

5. Round to two decimal places: According to the problem, we need to round our answer to two decimal places.

[tex]\[
x \approx 1.37
\][/tex]

Therefore, the solution to the equation [tex]\(3 \cdot e^x = 11.76\)[/tex] is [tex]\(x = 1.37\)[/tex].

So, the correct answer is:
C. [tex]\(x = 1.37\)[/tex]

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