Precalculus Examples

$\log_{3} (4) \log_{4} (5) \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Step 1

Rewrite $\log_{3} (4)$ using the change of base formula.

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Step 1.1

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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{4} (5) \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Step 1.2

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

$\frac{\log (4)}{\log (3)} \log_{4} (5) \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Step 2

Rewrite $\log_{4} (5)$ 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 $0$ .

$\frac{\log (4)}{\log (3)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Step 2.2

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

$\frac{\log (4)}{\log (3)} \cdot \frac{\log (5)}{\log (4)} \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Step 3

Cancel the common factor of $\log (4)$ .

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Step 3.1

Cancel the common factor.

$\frac{\log (4)}{\log (3)} \cdot \frac{\log (5)}{\log (4)} \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Step 3.2

Rewrite the expression.

$\frac{1}{\log (3)} \log (5) \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Step 4

Combine $\frac{1}{\log (3)}$ and $\log (5)$ .

$\frac{\log (5)}{\log (3)} \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Step 5

Rewrite $\log_{5} (6)$ using the change of base formula.

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Step 5.1

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 $0$ .

$\frac{\log (5)}{\log (3)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Step 5.2

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

$\frac{\log (5)}{\log (3)} \cdot \frac{\log (6)}{\log (5)} \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Step 6

Cancel the common factor of $\log (5)$ .

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Step 6.1

Cancel the common factor.

$\frac{\log (5)}{\log (3)} \cdot \frac{\log (6)}{\log (5)} \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Step 6.2

Rewrite the expression.

$\frac{1}{\log (3)} \log (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Step 7

Combine $\frac{1}{\log (3)}$ and $\log (6)$ .

$\frac{\log (6)}{\log (3)} \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Step 8

Rewrite $\log_{6} (7)$ using the change of base formula.

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Step 8.1

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 $0$ .

$\frac{\log (6)}{\log (3)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{7} (8) \log_{8} (9)$

Step 8.2

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

$\frac{\log (6)}{\log (3)} \cdot \frac{\log (7)}{\log (6)} \log_{7} (8) \log_{8} (9)$

Step 9

Cancel the common factor of $\log (6)$ .

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Step 9.1

Cancel the common factor.

$\frac{\log (6)}{\log (3)} \cdot \frac{\log (7)}{\log (6)} \log_{7} (8) \log_{8} (9)$

Step 9.2

Rewrite the expression.

$\frac{1}{\log (3)} \log (7) \log_{7} (8) \log_{8} (9)$

Step 10

Combine $\frac{1}{\log (3)}$ and $\log (7)$ .

$\frac{\log (7)}{\log (3)} \log_{7} (8) \log_{8} (9)$

Step 11

Rewrite $\log_{7} (8)$ using the change of base formula.

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Step 11.1

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 $0$ .

$\frac{\log (7)}{\log (3)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{8} (9)$

Step 11.2

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

$\frac{\log (7)}{\log (3)} \cdot \frac{\log (8)}{\log (7)} \log_{8} (9)$

Step 12

Cancel the common factor of $\log (7)$ .

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Step 12.1

Cancel the common factor.

$\frac{\log (7)}{\log (3)} \cdot \frac{\log (8)}{\log (7)} \log_{8} (9)$

Step 12.2

Rewrite the expression.

$\frac{1}{\log (3)} \log (8) \log_{8} (9)$

Step 13

Combine $\frac{1}{\log (3)}$ and $\log (8)$ .

$\frac{\log (8)}{\log (3)} \log_{8} (9)$

Step 14

Rewrite $\log_{8} (9)$ using the change of base formula.

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Step 14.1

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 $0$ .

$\frac{\log (8)}{\log (3)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 14.2

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

$\frac{\log (8)}{\log (3)} \cdot \frac{\log (9)}{\log (8)}$

Step 15

Cancel the common factor of $\log (8)$ .

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Step 15.1

Cancel the common factor.

$\frac{\log (8)}{\log (3)} \cdot \frac{\log (9)}{\log (8)}$

Step 15.2

Rewrite the expression.

$\frac{1}{\log (3)} \log (9)$

Step 16

Combine $\frac{1}{\log (3)}$ and $\log (9)$ .

$\frac{\log (9)}{\log (3)}$

Step 17

Rewrite $\log (9)$ as $\log (3^{2})$ .

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

Step 18

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

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

Step 19

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

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Step 19.1

Cancel the common factor.

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

Step 19.2

Divide $2$ by $1$ .

$2$