Write as a Single Logarithm 18(1/2 log of x+y-1/9 log of x-y)

$18 (\frac{1}{2} \log (x + y) - \frac{1}{9} \log (x - y))$

Step 1

Simplify each term.

Simplify $\frac{1}{2} \log (x + y)$ by moving $\frac{1}{2}$ inside the logarithm.

$18 (\log ({(x + y)}^{\frac{1}{2}}) - \frac{1}{9} \cdot \log (x - y))$

Multiply $- \frac{1}{9} \log (x - y)$ .

Reorder $\log (x - y)$ and $\frac{1}{9}$ .

$18 (\log ({(x + y)}^{\frac{1}{2}}) - (\frac{1}{9} \log (x - y)))$

Simplify $\frac{1}{9} \log (x - y)$ by moving $\frac{1}{9}$ inside the logarithm.

$18 (\log ({(x + y)}^{\frac{1}{2}}) - \log ({(x - y)}^{\frac{1}{9}}))$

Step 2

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

$18 \log (\frac{{(x + y)}^{\frac{1}{2}}}{{(x - y)}^{\frac{1}{9}}})$

Step 3

Simplify $18 \log (\frac{{(x + y)}^{\frac{1}{2}}}{{(x - y)}^{\frac{1}{9}}})$ by moving $18$ inside the logarithm.

$\log ({(\frac{{(x + y)}^{\frac{1}{2}}}{{(x - y)}^{\frac{1}{9}}})}^{18})$

Step 4

Apply the product rule to $\frac{{(x + y)}^{\frac{1}{2}}}{{(x - y)}^{\frac{1}{9}}}$ .

$\log (\frac{{({(x + y)}^{\frac{1}{2}})}^{18}}{{({(x - y)}^{\frac{1}{9}})}^{18}})$

Step 5

Multiply the exponents in ${({(x + y)}^{\frac{1}{2}})}^{18}$ .

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

$\log (\frac{{(x + y)}^{\frac{1}{2} \cdot 18}}{{({(x - y)}^{\frac{1}{9}})}^{18}})$

Cancel the common factor of $2$ .

Factor $2$ out of $18$ .

$\log (\frac{{(x + y)}^{\frac{1}{2} \cdot (2 (9))}}{{({(x - y)}^{\frac{1}{9}})}^{18}})$

Cancel the common factor.

$\log (\frac{{(x + y)}^{\frac{1}{2} \cdot (2 \cdot 9)}}{{({(x - y)}^{\frac{1}{9}})}^{18}})$

Rewrite the expression.

$\log (\frac{{(x + y)}^{9}}{{({(x - y)}^{\frac{1}{9}})}^{18}})$

Step 6

Multiply the exponents in ${({(x - y)}^{\frac{1}{9}})}^{18}$ .

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

$\log (\frac{{(x + y)}^{9}}{{(x - y)}^{\frac{1}{9} \cdot 18}})$

Cancel the common factor of $9$ .

Factor $9$ out of $18$ .

$\log (\frac{{(x + y)}^{9}}{{(x - y)}^{\frac{1}{9} \cdot (9 (2))}})$

Cancel the common factor.

$\log (\frac{{(x + y)}^{9}}{{(x - y)}^{\frac{1}{9} \cdot (9 \cdot 2)}})$

Rewrite the expression.

$\log (\frac{{(x + y)}^{9}}{{(x - y)}^{2}})$