Precalculus Examples

$\log_{3} (11 + 3) \log_{3} (2)$

Step 1

Add $11$ and $3$ .

$\log_{3} (14) \log_{3} (2)$

Step 2

Rewrite $\log_{3} (14)$ using the change of base formula.

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Step 2.1

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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{3} (2)$

Step 2.2

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

$\frac{\log (14)}{\log (3)} \log_{3} (2)$

Step 3

Rewrite $\log_{3} (2)$ 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 $0$ .

$\frac{\log (14)}{\log (3)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 3.2

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

$\frac{\log (14)}{\log (3)} \cdot \frac{\log (2)}{\log (3)}$

Step 4

Multiply $\frac{\log (14)}{\log (3)} \cdot \frac{\log (2)}{\log (3)}$ .

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Step 4.1

Multiply $\frac{\log (14)}{\log (3)}$ by $\frac{\log (2)}{\log (3)}$ .

$\frac{\log (14) \log (2)}{\log (3) \log (3)}$

Step 4.2

Raise $\log (3)$ to the power of $1$ .

$\frac{\log (14) \log (2)}{\log^{1} (3) \log (3)}$

Step 4.3

Raise $\log (3)$ to the power of $1$ .

$\frac{\log (14) \log (2)}{\log^{1} (3) \log^{1} (3)}$

Step 4.4

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

$\frac{\log (14) \log (2)}{{\log (3)}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$\frac{\log (14) \log (2)}{\log^{2} (3)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (14) \log (2)}{\log^{2} (3)}$

Decimal Form:

$1.51560273 \dots$