Precalculus Examples

$\frac{7 \log (3)}{2 \log (9) + \log (3)}$

Step 1

Simplify $7 \log (3)$ by moving $7$ inside the logarithm.

$\frac{\log (3^{7})}{2 \log (9) + \log (3)}$

Step 2

Simplify the denominator.

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Step 2.1

Simplify $2 \log (9)$ by moving $2$ inside the logarithm.

$\frac{\log (3^{7})}{\log (9^{2}) + \log (3)}$

Step 2.2

Raise $9$ to the power of $2$ .

$\frac{\log (3^{7})}{\log (81) + \log (3)}$

Step 2.3

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

$\frac{\log (3^{7})}{\log (81 \cdot 3)}$

Step 2.4

Multiply $81$ by $3$ .

$\frac{\log (3^{7})}{\log (243)}$

Step 3

Expand $\log (3^{7})$ by moving $7$ outside the logarithm.

$\frac{7 \log (3)}{\log (243)}$

Step 4

Rewrite $\log (243)$ as $\log (3^{5})$ .

$\frac{7 \log (3)}{\log (3^{5})}$

Step 5

Expand $\log (3^{5})$ by moving $5$ outside the logarithm.

$\frac{7 \log (3)}{5 \log (3)}$

Step 6

Cancel the common factor of $\log (3)$ .

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Step 6.1

Cancel the common factor.

$\frac{7 \log (3)}{5 \log (3)}$

Step 6.2

Rewrite the expression.

$\frac{7}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{5}$

Decimal Form:

$1.4$

Mixed Number Form:

$1 \frac{2}{5}$