QuestionSolve for $x$ . $4 \ln (x+3)=16$
Do not round any intermediate computations, and round your answer to the nearest hundredth. $x=$ $\square$

Studdy Solution

STEP 1

1. The equation involves a natural logarithm.
2. We will use properties of logarithms and algebraic manipulation to solve for $x$ .

STEP 2

1. Isolate the logarithmic expression.
2. Use the property of logarithms to eliminate the logarithm.
3. Solve for $x$ .

STEP 3

First, divide both sides of the equation by 4 to isolate the logarithmic expression:
$4 \ln(x+3) = 16$ $\ln(x+3) = \frac{16}{4}$ $\ln(x+3) = 4$

STEP 4

To eliminate the natural logarithm, exponentiate both sides using the base $e$ :
$\ln(x+3) = 4$ $e^{\ln(x+3)} = e^4$ $x+3 = e^4$

STEP 5

Solve for $x$ by subtracting 3 from both sides:
$x+3 = e^4$ $x = e^4 - 3$
Now, calculate $e^4$ and then subtract 3. Using a calculator, we find:
$e^4 \approx 54.5982$
Therefore:
$x = 54.5982 - 3$ $x \approx 51.5982$
Rounding to the nearest hundredth, we get:
$x \approx 51.60$
The value of $x$ is:
$\boxed{51.60}$

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