Questions: Solve the following equation for (x). [ e^7 x=12 ] (x=) □ (Type an exact answer.)

Solve the following equation for (x). [ e^7 x=12 ] (x=) □ (Type an exact answer.)
Transcript text: Solve the following equation for $x$. \[ e^{7 x}=12 \] $x=$ $\square$ (Type an exact answer.)
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Solution

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Solution Steps

Step 1: Take the Natural Logarithm

To solve the equation \( e^{7x} = 12 \), we first take the natural logarithm of both sides:

\[ \ln(e^{7x}) = \ln(12) \]

Step 2: Simplify Using Logarithmic Properties

Using the property of logarithms that states \( \ln(e^y) = y \), we simplify the left side:

\[ 7x = \ln(12) \]

Step 3: Solve for \( x \)

Next, we isolate \( x \) by dividing both sides by 7:

\[ x = \frac{\ln(12)}{7} \]

Calculating the value of \( \ln(12) \) gives approximately \( 2.4849 \). Thus, we have:

\[ x \approx \frac{2.4849}{7} \approx 0.3549 \]

Final Answer

\[ \boxed{x \approx 0.3549} \]

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