Simplify/Condense natural log of 40+2 natural log of 1/2+ natural log of x

$\ln (40) + 2 \ln (\frac{1}{2}) + \ln (x)$

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$2 \ln (\frac{1}{2}) + \ln (40 x)$

Simplify each term.

Simplify $2 \ln (\frac{1}{2})$ by moving $2$ inside the logarithm.

$\ln ({(\frac{1}{2})}^{2}) + \ln (40 x)$

Apply the product rule to $\frac{1}{2}$ .

$\ln (\frac{1^{2}}{2^{2}}) + \ln (40 x)$

One to any power is one.

$\ln (\frac{1}{2^{2}}) + \ln (40 x)$

Raise $2$ to the power of $2$ .

$\ln (\frac{1}{4}) + \ln (40 x)$

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (\frac{1}{4} (40 x))$

Cancel the common factor of $4$ .

Factor $4$ out of $40 x$ .

$\ln (\frac{1}{4} (4 (10 x)))$

Cancel the common factor.

$\ln (\frac{1}{4} (4 (10 x)))$

Rewrite the expression.

$\ln (10 x)$