Precalculus Examples

$5 \log (y) + \log (8 x + 5 y)$

Step 1

Simplify $5 \log (y)$ by moving $5$ inside the logarithm.

$\log (y^{5}) + \log (8 x + 5 y)$

Step 2

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

$\log (y^{5} (8 x + 5 y))$

Step 3

Apply the distributive property.

$\log (y^{5} (8 x) + y^{5} (5 y))$

Step 4

Reorder.

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Step 4.1

Rewrite using the commutative property of multiplication.

$\log (8 y^{5} x + y^{5} (5 y))$

Step 4.2

Rewrite using the commutative property of multiplication.

$\log (8 y^{5} x + 5 y^{5} y)$

Step 5

Multiply $y^{5}$ by $y$ by adding the exponents.

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Step 5.1

Move $y$ .

$\log (8 y^{5} x + 5 (y \cdot y^{5}))$

Step 5.2

Multiply $y$ by $y^{5}$ .

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Step 5.2.1

Raise $y$ to the power of $1$ .

$\log (8 y^{5} x + 5 (y^{1} y^{5}))$

Step 5.2.2

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

$\log (8 y^{5} x + 5 y^{1 + 5})$

Step 5.3

Add $1$ and $5$ .

$\log (8 y^{5} x + 5 y^{6})$