Question

solve for (x).

\\\ln (x + 2) + \ln 5 = \ln 4\\

(x = \square)

Explanation:

Apply product property of logarithms

We combine the terms on the left side using the product rule for logarithms.
Using the Logarithmic Equations knowledge point:

$$ \ln(a) + \ln(b) = \ln(a \cdot b) $$

$$ \ln(5(x + 2)) = \ln 4 $$

Equate the arguments

Since the natural logarithm function is one-to-one, we can set the arguments equal to each other.

$$ 5(x + 2) = 4 $$

Solve the linear equation

We distribute the 5 and solve for \(x\).

$$ 5x + 10 = 4 $$

$$ 5x = -6 $$

$$ x = -\frac{6}{5} $$

Verify the solution

We check if the solution is in the domain of the original logarithmic expressions.
For \(\ln(x+2)\) to be defined, we require:

$$ x + 2 > 0 \implies x > -2 $$

Since \(-\frac{6}{5} = -1.2 > -2\), the solution is valid.

Answer:

Solve for \(x\).

$$\ln (x + 2) + \ln 5 = \ln 4$$

\(x =\) <blank>\(-\frac{6}{5}\)</blank>