Question

Solve for the missing variable.

Your answer must be rounded to the nearest two decimal places. 2e3x = 32


0.92


1.04


2.48


6.88

Answer 1

To solve the equation \(2e^{3x} = 32\), we first isolate the exponential term:

1. Divide both sides by 2:
\[
e^{3x} = \frac{32}{2} = 16
\]

2. Take the natural logarithm of both sides:
\[
\ln(e^{3x}) = \ln(16)
\]

3. Using the property of logarithms that \(\ln(e^y) = y\):
\[
3x = \ln(16)
\]

4. Divide both sides by 3:
\[
x = \frac{\ln(16)}{3}
\]

5. Now, calculate \(\ln(16)\):
\[
\ln(16) = \ln(2^4) = 4\ln(2)
\]

Using \(\ln(2) \approx 0.693147\):
\[
\ln(16) \approx 4 \times 0.693147 \approx 2.772588
\]

6. Now, substitute this value back to find \(x\):
\[
x \approx \frac{2.772588}{3} \approx 0.924196
\]

7. Rounding to the nearest two decimal places:
\[
x \approx 0.92
\]

Thus, the answer is:
\[
\boxed{0.92}
\]