Simplify/Condense 5* log base 8 of c^3d^2+3* log base 8 of cd^7

$5 \cdot \log_{8} (c^{3} d^{2}) + 3 \cdot \log_{8} (c d^{7})$

Simplify each term.

Simplify $5 \log_{8} (c^{3} d^{2})$ by moving $5$ inside the logarithm.

$\log_{8} ({(c^{3} d^{2})}^{5}) + 3 \cdot \log_{8} (c d^{7})$

Apply the product rule to $c^{3} d^{2}$ .

$\log_{8} ({(c^{3})}^{5} {(d^{2})}^{5}) + 3 \cdot \log_{8} (c d^{7})$

Multiply the exponents in ${(c^{3})}^{5}$ .

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

$\log_{8} (c^{3 \cdot 5} {(d^{2})}^{5}) + 3 \cdot \log_{8} (c d^{7})$

Multiply $3$ by $5$ .

$\log_{8} (c^{15} {(d^{2})}^{5}) + 3 \cdot \log_{8} (c d^{7})$

Multiply the exponents in ${(d^{2})}^{5}$ .

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

$\log_{8} (c^{15} d^{2 \cdot 5}) + 3 \cdot \log_{8} (c d^{7})$

Multiply $2$ by $5$ .

$\log_{8} (c^{15} d^{10}) + 3 \cdot \log_{8} (c d^{7})$

Simplify $3 \log_{8} (c d^{7})$ by moving $3$ inside the logarithm.

$\log_{8} (c^{15} d^{10}) + \log_{8} ({(c d^{7})}^{3})$

Apply the product rule to $c d^{7}$ .

$\log_{8} (c^{15} d^{10}) + \log_{8} (c^{3} {(d^{7})}^{3})$

Multiply the exponents in ${(d^{7})}^{3}$ .

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

$\log_{8} (c^{15} d^{10}) + \log_{8} (c^{3} d^{7 \cdot 3})$

Multiply $7$ by $3$ .

$\log_{8} (c^{15} d^{10}) + \log_{8} (c^{3} d^{21})$

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

$\log_{8} (c^{15} d^{10} (c^{3} d^{21}))$

Multiply $c^{15}$ by $c^{3}$ by adding the exponents.

Move $c^{3}$ .

$\log_{8} (c^{3} c^{15} d^{10} d^{21})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{8} (c^{3 + 15} d^{10} d^{21})$

Add $3$ and $15$ .

$\log_{8} (c^{18} d^{10} d^{21})$

Multiply $d^{10}$ by $d^{21}$ by adding the exponents.

Move $d^{21}$ .

$\log_{8} (c^{18} (d^{21} d^{10}))$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{8} (c^{18} d^{21 + 10})$

Add $21$ and $10$ .

$\log_{8} (c^{18} d^{31})$