Precalculus Examples

$\log_{729} (9) + \log_{27} (9) + \log_{3} (9)$

Step 1

Simplify each term.

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Step 1.1

Logarithm base $729$ of $9$ is $\frac{1}{3}$ .

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Step 1.1.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{6} + \log_{27} (9) + \log_{3} (9)$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 1}{2 \cdot 3} + \log_{27} (9) + \log_{3} (9)$

Step 1.1.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 3} + \log_{27} (9) + \log_{3} (9)$

Step 1.1.2.3

Rewrite the expression.

$\frac{1}{3} + \log_{27} (9) + \log_{3} (9)$

Step 1.2

Logarithm base $27$ of $9$ is $\frac{2}{3}$ .

$\frac{1}{3} + \frac{2}{3} + \log_{3} (9)$

Step 1.3

Logarithm base $3$ of $9$ is $2$ .

$\frac{1}{3} + \frac{2}{3} + 2$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

$2 + \frac{1 + 2}{3}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Add $1$ and $2$ .

$2 + \frac{3}{3}$

Step 2.2.2

Divide $3$ by $3$ .

$2 + 1$

Step 2.2.3

Add $2$ and $1$ .

$3$