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Answer :
To solve the equation [tex]\(3 \cdot e^x = 11.76\)[/tex], we need to isolate [tex]\(x\)[/tex]. Here’s a step-by-step explanation:
1. Divide by 3: Start by dividing both sides of the equation by 3 to isolate the exponential term:
[tex]\[
e^x = \frac{11.76}{3}
\][/tex]
[tex]\[
e^x = 3.92
\][/tex]
2. Apply the natural logarithm: To solve for [tex]\(x\)[/tex], we take the natural logarithm (ln) of both sides. The natural logarithm is the inverse operation of the exponential function, which helps us solve for the exponent:
[tex]\[
\ln(e^x) = \ln(3.92)
\][/tex]
With the property of logarithms [tex]\(\ln(e^x) = x\)[/tex], the equation simplifies to:
[tex]\[
x = \ln(3.92)
\][/tex]
3. Calculate the natural logarithm: Calculate [tex]\(\ln(3.92)\)[/tex]. This will give us the value of [tex]\(x\)[/tex].
4. Round the answer: The result of [tex]\(\ln(3.92)\)[/tex] is approximately 1.3661. Rounding this to two decimal places gives:
[tex]\[
x \approx 1.37
\][/tex]
Therefore, the solution to the equation [tex]\(3 \cdot e^x = 11.76\)[/tex], rounded to two decimal places, is [tex]\(x = 1.37\)[/tex]. The correct option is D. [tex]\(x = 1.37\)[/tex].
1. Divide by 3: Start by dividing both sides of the equation by 3 to isolate the exponential term:
[tex]\[
e^x = \frac{11.76}{3}
\][/tex]
[tex]\[
e^x = 3.92
\][/tex]
2. Apply the natural logarithm: To solve for [tex]\(x\)[/tex], we take the natural logarithm (ln) of both sides. The natural logarithm is the inverse operation of the exponential function, which helps us solve for the exponent:
[tex]\[
\ln(e^x) = \ln(3.92)
\][/tex]
With the property of logarithms [tex]\(\ln(e^x) = x\)[/tex], the equation simplifies to:
[tex]\[
x = \ln(3.92)
\][/tex]
3. Calculate the natural logarithm: Calculate [tex]\(\ln(3.92)\)[/tex]. This will give us the value of [tex]\(x\)[/tex].
4. Round the answer: The result of [tex]\(\ln(3.92)\)[/tex] is approximately 1.3661. Rounding this to two decimal places gives:
[tex]\[
x \approx 1.37
\][/tex]
Therefore, the solution to the equation [tex]\(3 \cdot e^x = 11.76\)[/tex], rounded to two decimal places, is [tex]\(x = 1.37\)[/tex]. The correct option is D. [tex]\(x = 1.37\)[/tex].
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