Question

solve the logarithmic equation. be sure to reject any value of x that is not in the domain of the original logarithmic expression.
4 ln(6x) = 12
solve the equation. what is the exact solution? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is
(type an exact answer in simplified form. use integers or fractions for any numbers in the expression)
b. there are infinitely many solutions.
c. there is no solution
what is the decimal approximation to the solution? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is
(type an integer or decimal rounded to two decimal places as needed)
b. there are infinitely many solutions.
c. there is no solution.

Explanation:

Step1: Isolate the logarithm

We start with the equation \(4\ln(6x) = 12\). First, divide both sides by 4 to isolate the natural logarithm term.

$$ \frac{4\ln(6x)}{4}=\frac{12}{4} $$

Simplifying both sides, we get \(\ln(6x)=3\).

Step2: Convert to exponential form

Recall that if \(\ln(a)=b\), then \(a = e^{b}\) (since the natural logarithm has base \(e\)). Applying this to \(\ln(6x)=3\), we convert it to exponential form:

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

Step3: Solve for \(x\)

Now, divide both sides by 6 to solve for \(x\):

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

We can also find the decimal approximation. We know that \(e^{3}\approx20.0855\), so:

$$ x=\frac{20.0855}{6}\approx3.35 $$

Answer:

(Exact Solution):
A. The solution set is \(\frac{e^{3}}{6}\)