Precalculus Examples

$\log_{3} (x - 2) + 5 \log_{3} (x + 7)$

Step 1

Simplify $5 \log_{3} (x + 7)$ by moving $5$ inside the logarithm.

$\log_{3} (x - 2) + \log_{3} ({(x + 7)}^{5})$

Step 2

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

$\log_{3} ((x - 2) {(x + 7)}^{5})$

Step 3

Apply the distributive property.

$\log_{3} (x {(x + 7)}^{5} - 2 {(x + 7)}^{5})$

Step 4

Rewrite $\log_{3} (x {(x + 7)}^{5} - 2 {(x + 7)}^{5})$ using the change of base formula.

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Step 4.1

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 4.2

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

$\frac{\log (x {(x + 7)}^{5} - 2 {(x + 7)}^{5})}{\log (3)}$