Write as a Single Logarithm 3 natural log of a-1/2( natural log of b+ natural log of c^2)

$3 \ln (a) - \frac{1}{2} (\ln (b) + \ln (c^{2}))$

Step 1

Simplify each term.

Simplify $3 \ln (a)$ by moving $3$ inside the logarithm.

$\ln (a^{3}) - \frac{1}{2} \cdot (\ln (b) + \ln (c^{2}))$

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

$\ln (a^{3}) - \frac{1}{2} \cdot \ln (b c^{2})$

Multiply $- \frac{1}{2} \ln (b c^{2})$ .

Reorder $\ln (b c^{2})$ and $\frac{1}{2}$ .

$\ln (a^{3}) - (\frac{1}{2} \ln (b c^{2}))$

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

$\ln (a^{3}) - \ln ({(b c^{2})}^{\frac{1}{2}})$

Apply the product rule to $b c^{2}$ .

$\ln (a^{3}) - \ln (b^{\frac{1}{2}} {(c^{2})}^{\frac{1}{2}})$

Multiply the exponents in ${(c^{2})}^{\frac{1}{2}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\ln (a^{3}) - \ln (b^{\frac{1}{2}} c^{2 (\frac{1}{2})})$

Cancel the common factor of $2$ .

Cancel the common factor.

$\ln (a^{3}) - \ln (b^{\frac{1}{2}} c^{2 (\frac{1}{2})})$

Rewrite the expression.

$\ln (a^{3}) - \ln (b^{\frac{1}{2}} c^{1})$

Simplify.

$\ln (a^{3}) - \ln (b^{\frac{1}{2}} c)$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{a^{3}}{b^{\frac{1}{2}} c})$

Step 3

Reorder factors in $\ln (\frac{a^{3}}{b^{\frac{1}{2}} c})$ .

$\ln (\frac{a^{3}}{c b^{\frac{1}{2}}})$