College

If [tex]4^{3x} = 37.9[/tex], what is the value of [tex]x[/tex]?

A. [tex]x \approx 2.622[/tex]
B. [tex]x \approx 0.326[/tex]
C. [tex]x \approx 0.874[/tex]
D. [tex]x \approx 7.866[/tex]

Answer :

To solve the equation [tex]\(4^{3x} = 37.9\)[/tex] and find the value of [tex]\(x\)[/tex], you can use logarithms. Here's how:

1. Take the logarithm of both sides:
Start by taking the natural logarithm (ln) of both sides of the equation to bring down the exponent.

[tex]\[
\ln(4^{3x}) = \ln(37.9)
\][/tex]

2. Apply the power rule of logarithms:
The power rule states that [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]. Use this rule to simplify the left-hand side of the equation.

[tex]\[
3x \cdot \ln(4) = \ln(37.9)
\][/tex]

3. Isolate [tex]\(x\)[/tex]:
Solve for [tex]\(x\)[/tex] by dividing both sides by [tex]\(3 \cdot \ln(4)\)[/tex].

[tex]\[
x = \frac{\ln(37.9)}{3 \cdot \ln(4)}
\][/tex]

4. Calculate the values:

- Find [tex]\(\ln(4)\)[/tex] and [tex]\(\ln(37.9)\)[/tex] using a calculator.

- Insert these values into the equation to find [tex]\(x\)[/tex]:

[tex]\[
x \approx \frac{3.634951112088381}{3 \times 1.3862943611198906}
\][/tex]

- Simplify the calculation to find the approximate value of [tex]\(x\)[/tex].

[tex]\[
x \approx 0.874
\][/tex]

Thus, the value of [tex]\(x\)[/tex] is approximately 0.874.

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