Simplify/Condense 1/2( log base 6 of k^9- log base 6 of k^3)

$\frac{1}{2} (\log_{6} (k^{9}) - \log_{6} (k^{3}))$

Step 1

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

$\frac{1}{2} \cdot \log_{6} (\frac{k^{9}}{k^{3}})$

Step 2

Cancel the common factor of $k^{9}$ and $k^{3}$ .

Factor $k^{3}$ out of $k^{9}$ .

$\frac{1}{2} \cdot \log_{6} (\frac{k^{3} k^{6}}{k^{3}})$

Cancel the common factors.

Multiply by $1$ .

$\frac{1}{2} \cdot \log_{6} (\frac{k^{3} k^{6}}{k^{3} \cdot 1})$

Cancel the common factor.

$\frac{1}{2} \cdot \log_{6} (\frac{k^{3} k^{6}}{k^{3} \cdot 1})$

Rewrite the expression.

$\frac{1}{2} \cdot \log_{6} (\frac{k^{6}}{1})$

Divide $k^{6}$ by $1$ .

$\frac{1}{2} \cdot \log_{6} (k^{6})$

Step 3

Simplify $\frac{1}{2} \log_{6} (k^{6})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{6} ({(k^{6})}^{\frac{1}{2}})$

Step 4

Multiply the exponents in ${(k^{6})}^{\frac{1}{2}}$ .

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

$\log_{6} (k^{6 (\frac{1}{2})})$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$\log_{6} (k^{2 (3) \frac{1}{2}})$

Cancel the common factor.

$\log_{6} (k^{2 \cdot 3 \frac{1}{2}})$

Rewrite the expression.

$\log_{6} (k^{3})$