Precalculus Examples

$12 \log_{7} (3) + 12 \log_{7} (11) - 3 \log_{7} (5)$

Step 1

Simplify each term.

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Step 1.1

Simplify $12 \log_{7} (3)$ by moving $12$ inside the logarithm.

$\log_{7} (3^{12}) + 12 \log_{7} (11) - 3 \log_{7} (5)$

Step 1.2

Raise $3$ to the power of $12$ .

$\log_{7} (531441) + 12 \log_{7} (11) - 3 \log_{7} (5)$

Step 1.3

Simplify $12 \log_{7} (11)$ by moving $12$ inside the logarithm.

$\log_{7} (531441) + \log_{7} (11^{12}) - 3 \log_{7} (5)$

Step 1.4

Raise $11$ to the power of $12$ .

$\log_{7} (531441) + \log_{7} (3138428376721) - 3 \log_{7} (5)$

Step 1.5

Simplify $- 3 \log_{7} (5)$ by moving $3$ inside the logarithm.

$\log_{7} (531441) + \log_{7} (3138428376721) - \log_{7} (5^{3})$

Step 1.6

Raise $5$ to the power of $3$ .

$\log_{7} (531441) + \log_{7} (3138428376721) - \log_{7} (125)$

Step 2

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

$\log_{7} (531441 \cdot 3138428376721) - \log_{7} (125)$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{7} (\frac{531441 \cdot 3138428376721}{125})$

Step 4

Multiply $531441$ by $3138428376721$ .

$\log_{7} (\frac{1667889514952984961}{125})$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\log_{7} (\frac{1667889514952984961}{125})$

Decimal Form:

$19.08093085 \dots$