Simplify/Condense 4* log base 7 of a+(5* log base 7 of a-3* log base 7 of b)

$4 \cdot \log_{7} (a) + (5 \cdot \log_{7} (a) - 3 \cdot \log_{7} (b))$

Step 1

Remove parentheses.

$4 \cdot \log_{7} (a) + 5 \cdot \log_{7} (a) - 3 \cdot \log_{7} (b)$

Step 2

Simplify each term.

Simplify $4 \log_{7} (a)$ by moving $4$ inside the logarithm.

$\log_{7} (a^{4}) + 5 \cdot \log_{7} (a) - 3 \cdot \log_{7} (b)$

Simplify $5 \log_{7} (a)$ by moving $5$ inside the logarithm.

$\log_{7} (a^{4}) + \log_{7} (a^{5}) - 3 \cdot \log_{7} (b)$

Simplify $- 3 \log_{7} (b)$ by moving $3$ inside the logarithm.

$\log_{7} (a^{4}) + \log_{7} (a^{5}) - \log_{7} (b^{3})$

Step 3

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

$\log_{7} (a^{4} a^{5}) - \log_{7} (b^{3})$

Step 4

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

$\log_{7} (\frac{a^{4} a^{5}}{b^{3}})$

Step 5

Multiply $a^{4}$ by $a^{5}$ by adding the exponents.

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

$\log_{7} (\frac{a^{4 + 5}}{b^{3}})$

Add $4$ and $5$ .

$\log_{7} (\frac{a^{9}}{b^{3}})$