Precalculus Examples

$\frac{\log_{6} (5)}{3} + \frac{\log_{6} (8)}{6} + \frac{\log_{6} (11)}{3}$

Step 1

Combine the numerators over the common denominator.

$\frac{\log_{6} (5) + \log_{6} (11)}{3} + \frac{\log_{6} (8)}{6}$

Step 2

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

$\frac{\log_{6} (5 \cdot 11)}{3} + \frac{\log_{6} (8)}{6}$

Step 3

Multiply $5$ by $11$ .

$\frac{\log_{6} (55)}{3} + \frac{\log_{6} (8)}{6}$

Step 4

Simplify each term.

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Step 4.1

Rewrite $\frac{\log_{6} (55)}{3}$ as $\frac{1}{3} \log_{6} (55)$ .

$\frac{1}{3} \log_{6} (55) + \frac{\log_{6} (8)}{6}$

Step 4.2

Simplify $\frac{1}{3} \log_{6} (55)$ by moving $\frac{1}{3}$ inside the logarithm.

$\log_{6} (55^{\frac{1}{3}}) + \frac{\log_{6} (8)}{6}$

Step 4.3

Rewrite $\log_{6} (8)$ as $\log_{6} (2^{3})$ .

$\log_{6} (55^{\frac{1}{3}}) + \frac{\log_{6} (2^{3})}{6}$

Step 4.4

Expand $\log_{6} (2^{3})$ by moving $3$ outside the logarithm.

$\log_{6} (55^{\frac{1}{3}}) + \frac{3 \log_{6} (2)}{6}$

Step 4.5

Cancel the common factor of $3$ and $6$ .

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Step 4.5.1

Factor $3$ out of $3 \log_{6} (2)$ .

$\log_{6} (55^{\frac{1}{3}}) + \frac{3 (\log_{6} (2))}{6}$

Step 4.5.2

Cancel the common factors.

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Step 4.5.2.1

Factor $3$ out of $6$ .

$\log_{6} (55^{\frac{1}{3}}) + \frac{3 \log_{6} (2)}{3 \cdot 2}$

Step 4.5.2.2

Cancel the common factor.

$\log_{6} (55^{\frac{1}{3}}) + \frac{3 \log_{6} (2)}{3 \cdot 2}$

Step 4.5.2.3

Rewrite the expression.

$\log_{6} (55^{\frac{1}{3}}) + \frac{\log_{6} (2)}{2}$

Step 4.6

Rewrite $\frac{\log_{6} (2)}{2}$ as $\frac{1}{2} \log_{6} (2)$ .

$\log_{6} (55^{\frac{1}{3}}) + \frac{1}{2} \log_{6} (2)$

Step 4.7

Simplify $\frac{1}{2} \log_{6} (2)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{6} (55^{\frac{1}{3}}) + \log_{6} (2^{\frac{1}{2}})$

Step 5

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

$\log_{6} (55^{\frac{1}{3}} \cdot 2^{\frac{1}{2}})$

Step 6

Rewrite $\log_{6} (55^{\frac{1}{3}} \cdot 2^{\frac{1}{2}})$ using the change of base formula.

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Step 6.1

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 6.2

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

$\frac{\log (55^{\frac{1}{3}} \cdot 2^{\frac{1}{2}})}{\log (6)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (55^{\frac{1}{3}} \cdot 2^{\frac{1}{2}})}{\log (6)}$

Decimal Form:

$0.93893814 \dots$