Algebra Examples

$3 \log_{6} (x) + 5 \log_{6} (x - 6)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log_{6} (x)$ by moving $3$ inside the logarithm.

$\log_{6} (x^{3}) + 5 \log_{6} (x - 6)$

Step 1.2

Simplify $5 \log_{6} (x - 6)$ by moving $5$ inside the logarithm.

$\log_{6} (x^{3}) + \log_{6} ({(x - 6)}^{5})$

Step 2

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

$\log_{6} (x^{3} {(x - 6)}^{5})$

Step 3

Rewrite $\log_{6} (x^{3} {(x - 6)}^{5})$ using the change of base formula.

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Step 3.1

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 3.2

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

$\frac{\log (x^{3} {(x - 6)}^{5})}{\log (6)}$