Precalculus Examples

$\log_{2} (3) \cdot \log_{3} (4) \cdot \log_{4} (5) \cdot \log_{5} (6) \cdot \log_{6} (7) \cdot \log_{7} (8) \cdot \log_{8} (9)$

Multiply $\log_{2} (3) \log_{3} (4) \log_{4} (5) \log_{5} (6) \log_{6} (7) \log_{7} (8)$ by $\log_{8} (9)$ .

$\log_{2} (3) \log_{3} (4) \log_{4} (5) \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Rewrite $\log_{2} (3)$ using the change of base formula.

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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 $0$ .

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

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

$\frac{\log (3)}{\log (2)} \log_{3} (4) \log_{4} (5) \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

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

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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 $0$ .

$\frac{\log (3)}{\log (2)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{4} (5) \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

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

$\frac{\log (3)}{\log (2)} \cdot \frac{\log (4)}{\log (3)} \log_{4} (5) \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

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

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Cancel the common factor.

$\frac{\log (3)}{\log (2)} \cdot \frac{\log (4)}{\log (3)} \log_{4} (5) \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Rewrite the expression.

$\frac{1}{\log (2)} \log (4) \log_{4} (5) \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Combine $\frac{1}{\log (2)}$ and $\log (4)$ .

$\frac{\log (4)}{\log (2)} \log_{4} (5) \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Rewrite $\log (4)$ as $\log (2^{2})$ .

$\frac{\log (2^{2})}{\log (2)} \log_{4} (5) \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

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

$\frac{2 \log (2)}{\log (2)} \log_{4} (5) \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Cancel the common factor of $\log (2)$ .

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Cancel the common factor.

$\frac{2 \log (2)}{\log (2)} \log_{4} (5) \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Divide $2$ by $1$ .

$2 \log_{4} (5) \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Rewrite $\log_{4} (5)$ using the change of base formula.

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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 $0$ .

$2 \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

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

$2 \frac{\log (5)}{\log (4)} \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Multiply $2 \frac{\log (5)}{\log (4)}$ .

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Combine $2$ and $\frac{\log (5)}{\log (4)}$ .

$\frac{2 \log (5)}{\log (4)} \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Simplify $2 \log (5)$ by moving $2$ inside the logarithm.

$\frac{\log (5^{2})}{\log (4)} \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Raise $5$ to the power of $2$ .

$\frac{\log (25)}{\log (4)} \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Rewrite $\log (25)$ as $\log (5^{2})$ .

$\frac{\log (5^{2})}{\log (4)} \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Expand $\log (5^{2})$ by moving $2$ outside the logarithm.

$\frac{2 \log (5)}{\log (4)} \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Rewrite $\log (4)$ as $\log (2^{2})$ .

$\frac{2 \log (5)}{\log (2^{2})} \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

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

$\frac{2 \log (5)}{2 \log (2)} \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 \log (5)}{2 \log (2)} \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Rewrite the expression.

$\frac{\log (5)}{\log (2)} \log_{5} (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

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

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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 $0$ .

$\frac{\log (5)}{\log (2)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{6} (7) \log_{7} (8) \log_{8} (9)$

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

$\frac{\log (5)}{\log (2)} \cdot \frac{\log (6)}{\log (5)} \log_{6} (7) \log_{7} (8) \log_{8} (9)$

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

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Cancel the common factor.

$\frac{\log (5)}{\log (2)} \cdot \frac{\log (6)}{\log (5)} \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Rewrite the expression.

$\frac{1}{\log (2)} \log (6) \log_{6} (7) \log_{7} (8) \log_{8} (9)$

Combine $\frac{1}{\log (2)}$ and $\log (6)$ .

$\frac{\log (6)}{\log (2)} \log_{6} (7) \log_{7} (8) \log_{8} (9)$

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

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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 $0$ .

$\frac{\log (6)}{\log (2)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{7} (8) \log_{8} (9)$

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

$\frac{\log (6)}{\log (2)} \cdot \frac{\log (7)}{\log (6)} \log_{7} (8) \log_{8} (9)$

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

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Cancel the common factor.

$\frac{\log (6)}{\log (2)} \cdot \frac{\log (7)}{\log (6)} \log_{7} (8) \log_{8} (9)$

Rewrite the expression.

$\frac{1}{\log (2)} \log (7) \log_{7} (8) \log_{8} (9)$

Combine $\frac{1}{\log (2)}$ and $\log (7)$ .

$\frac{\log (7)}{\log (2)} \log_{7} (8) \log_{8} (9)$

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

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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 $0$ .

$\frac{\log (7)}{\log (2)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{8} (9)$

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

$\frac{\log (7)}{\log (2)} \cdot \frac{\log (8)}{\log (7)} \log_{8} (9)$

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

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Cancel the common factor.

$\frac{\log (7)}{\log (2)} \cdot \frac{\log (8)}{\log (7)} \log_{8} (9)$

Rewrite the expression.

$\frac{1}{\log (2)} \log (8) \log_{8} (9)$

Combine $\frac{1}{\log (2)}$ and $\log (8)$ .

$\frac{\log (8)}{\log (2)} \log_{8} (9)$

Rewrite $\log (8)$ as $\log (2^{3})$ .

$\frac{\log (2^{3})}{\log (2)} \log_{8} (9)$

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

$\frac{3 \log (2)}{\log (2)} \log_{8} (9)$

Cancel the common factor of $\log (2)$ .

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Cancel the common factor.

$\frac{3 \log (2)}{\log (2)} \log_{8} (9)$

Divide $3$ by $1$ .

$3 \log_{8} (9)$

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

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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 $0$ .

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

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

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

Multiply $3 \frac{\log (9)}{\log (8)}$ .

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Combine $3$ and $\frac{\log (9)}{\log (8)}$ .

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

Simplify $3 \log (9)$ by moving $3$ inside the logarithm.

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

Raise $9$ to the power of $3$ .

$\frac{\log (729)}{\log (8)}$

Rewrite $\log (729)$ as $\log (3^{6})$ .

$\frac{\log (3^{6})}{\log (8)}$

Expand $\log (3^{6})$ by moving $6$ outside the logarithm.

$\frac{6 \log (3)}{\log (8)}$

Rewrite $\log (8)$ as $\log (2^{3})$ .

$\frac{6 \log (3)}{\log (2^{3})}$

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

$\frac{6 \log (3)}{3 \log (2)}$

Cancel the common factor of $6$ and $3$ .

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Factor $3$ out of $6 \log (3)$ .

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

Cancel the common factors.

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Factor $3$ out of $3 \log (2)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify $2 \log (3)$ by moving $2$ inside the logarithm.

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

Raise $3$ to the power of $2$ .

$\frac{\log (9)}{\log (2)}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (9)}{\log (2)}$

Decimal Form:

$3.16992500 \dots$