Question

section 1.8: logarithmic functions (homewor
score: 70/120 answered: 7/12
question 8
if $e^{3x} = 16$, then $x = $
question help: video

Explanation:

Step1: Take natural log of both sides

To solve for \( x \) in the equation \( e^{3x} = 16 \), we start by taking the natural logarithm (ln) of both sides. Recall that \( \ln(e^y)=y \) due to the inverse relationship between the exponential function with base \( e \) and the natural logarithm. So, applying \( \ln \) to both sides gives:

$$ \ln(e^{3x})=\ln(16) $$

Using the property \( \ln(e^y) = y \), the left - hand side simplifies to \( 3x \). So we have:

$$ 3x=\ln(16) $$

Step2: Solve for x

Now, we want to isolate \( x \). We can do this by dividing both sides of the equation \( 3x=\ln(16) \) by 3.

$$ x = \frac{\ln(16)}{3} $$

We can also simplify \( \ln(16) \) since \( 16 = 2^4 \), and using the property \( \ln(a^b)=b\ln(a) \), we have \( \ln(16)=\ln(2^4) = 4\ln(2) \). So \( x=\frac{4\ln(2)}{3}\approx\frac{4\times0.6931}{3}=\frac{2.7724}{3}\approx0.9241 \), but the exact form is \( \frac{\ln(16)}{3} \) or \( \frac{4\ln(2)}{3} \).

Answer:

\( \frac{\ln(16)}{3} \) (or \( \frac{4\ln(2)}{3} \) or the approximate decimal value depending on the requirement)