Precalculus Examples

$\log_{4} (12 x) + \log_{4} (2 x^{7})$

Step 1

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

$\log_{4} (12 x (2 x^{7}))$

Step 2

Rewrite using the commutative property of multiplication.

$\log_{4} (12 \cdot 2 x \cdot x^{7})$

Step 3

Multiply $x$ by $x^{7}$ by adding the exponents.

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Step 3.1

Move $x^{7}$ .

$\log_{4} (12 \cdot 2 (x^{7} x))$

Step 3.2

Multiply $x^{7}$ by $x$ .

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Step 3.2.1

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

$\log_{4} (12 \cdot 2 (x^{7} x^{1}))$

Step 3.2.2

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

$\log_{4} (12 \cdot 2 x^{7 + 1})$

Step 3.3

Add $7$ and $1$ .

$\log_{4} (12 \cdot 2 x^{8})$

Step 4

Multiply $12$ by $2$ .

$\log_{4} (24 x^{8})$

Step 5

Rewrite $\log_{4} (24 x^{8})$ using the change of base formula.

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Step 5.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 5.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (24 x^{8})}{\log (4)}$