We need to solve the equation $\ln(x) + \ln(6) = 3$ for $x$, and give the answer correct to 3 decimal places.

1. Problem Description

We need to solve the equation $\ln(x) + \ln(6) = 3$ for $x$, and give the answer correct to 3 decimal places.

2. Solution Steps

We can use the property of logarithms that $\ln(a) + \ln(b) = \ln(ab)$. Therefore,

$\ln(x) + \ln(6) = \ln(6x)$

So, the equation becomes $\ln(6x) = 3$.

To solve for $x$, we can exponentiate both sides with base $e$:

$e^{\ln(6x)} = e^3$

$6x = e^3$

Now, we can solve for $x$ by dividing both sides by 6:

$x = \frac{e^3}{6}$

We can approximate the value of $e^3$ as 20.

0

8

5

5. $x = \frac{20.0855}{6}$

$x \approx 3.34758$

Rounding to 3 decimal places, we have $x \approx 3.348$.

3. Final Answer

$x = \frac{e^3}{6} \approx 3.348$