Precalculus Examples

$(\log_{27} (36)) (\log_{36} (49)) (\log_{49} (81))$

Step 1

Rewrite $\log_{27} (36)$ as $\log_{27} (2^{2} \cdot 3^{2})$ .

$\log_{27} (2^{2} \cdot 3^{2}) \log_{36} (49) \log_{49} (81)$

Step 2

Rewrite $\log_{27} (2^{2} \cdot 3^{2})$ as $\log_{27} (2^{2}) + \log_{27} (3^{2})$ .

$(\log_{27} (2^{2}) + \log_{27} (3^{2})) \log_{36} (49) \log_{49} (81)$

Step 3

Simplify each term.

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Step 3.1

Expand $\log_{27} (2^{2})$ by moving $2$ outside the logarithm.

$(2 \log_{27} (2) + \log_{27} (3^{2})) \log_{36} (49) \log_{49} (81)$

Step 3.2

Expand $\log_{27} (3^{2})$ by moving $2$ outside the logarithm.

$(2 \log_{27} (2) + 2 \log_{27} (3)) \log_{36} (49) \log_{49} (81)$

Step 3.3

Logarithm base $27$ of $3$ is $\frac{1}{3}$ .

$(2 \log_{27} (2) + 2 (\frac{1}{3})) \log_{36} (49) \log_{49} (81)$

Step 3.4

Combine $2$ and $\frac{1}{3}$ .

$(2 \log_{27} (2) + \frac{2}{3}) \log_{36} (49) \log_{49} (81)$

Step 4

Rewrite $\log_{36} (49)$ as $\log_{36} (7^{2})$ .

$(2 \log_{27} (2) + \frac{2}{3}) \log_{36} (7^{2}) \log_{49} (81)$

Step 5

Expand $\log_{36} (7^{2})$ by moving $2$ outside the logarithm.

$(2 \log_{27} (2) + \frac{2}{3}) (2 \log_{36} (7)) \log_{49} (81)$

Step 6

Apply the distributive property.

$(2 \log_{27} (2) (2 \log_{36} (7)) + \frac{2}{3} (2 \log_{36} (7))) \log_{49} (81)$

Step 7

Multiply $2$ by $2$ .

$(4 \log_{27} (2) \log_{36} (7) + \frac{2}{3} (2 \log_{36} (7))) \log_{49} (81)$

Step 8

Multiply $\frac{2}{3} (2 \log_{36} (7))$ .

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Step 8.1

Combine $2$ and $\frac{2}{3}$ .

$(4 \log_{27} (2) \log_{36} (7) + \frac{2 \cdot 2}{3} \log_{36} (7)) \log_{49} (81)$

Step 8.2

Multiply $2$ by $2$ .

$(4 \log_{27} (2) \log_{36} (7) + \frac{4}{3} \log_{36} (7)) \log_{49} (81)$

Step 8.3

Combine $\frac{4}{3}$ and $\log_{36} (7)$ .

$(4 \log_{27} (2) \log_{36} (7) + \frac{4 \log_{36} (7)}{3}) \log_{49} (81)$

Step 9

Rewrite $\log_{49} (81)$ as $\log_{49} (3^{4})$ .

$(4 \log_{27} (2) \log_{36} (7) + \frac{4 \log_{36} (7)}{3}) \log_{49} (3^{4})$

Step 10

Expand $\log_{49} (3^{4})$ by moving $4$ outside the logarithm.

$(4 \log_{27} (2) \log_{36} (7) + \frac{4 \log_{36} (7)}{3}) (4 \log_{49} (3))$

Step 11

Apply the distributive property.

$4 \log_{27} (2) \log_{36} (7) (4 \log_{49} (3)) + \frac{4 \log_{36} (7)}{3} (4 \log_{49} (3))$

Step 12

Multiply $4$ by $4$ .

$16 \log_{27} (2) \log_{36} (7) \log_{49} (3) + \frac{4 \log_{36} (7)}{3} (4 \log_{49} (3))$

Step 13

Multiply $\frac{4 \log_{36} (7)}{3} (4 \log_{49} (3))$ .

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Step 13.1

Combine $4$ and $\frac{4 \log_{36} (7)}{3}$ .

$16 \log_{27} (2) \log_{36} (7) \log_{49} (3) + \frac{4 (4 \log_{36} (7))}{3} \log_{49} (3)$

Step 13.2

Multiply $4$ by $4$ .

$16 \log_{27} (2) \log_{36} (7) \log_{49} (3) + \frac{16 \log_{36} (7)}{3} \log_{49} (3)$

Step 13.3

Combine $\frac{16 \log_{36} (7)}{3}$ and $\log_{49} (3)$ .

$16 \log_{27} (2) \log_{36} (7) \log_{49} (3) + \frac{16 \log_{36} (7) \log_{49} (3)}{3}$