Precalculus Examples

$\log_{2} (5) \cdot \log_{5} (6) \cdot \log_{6} (8)$

Step 1

Multiply $\log_{2} (5) \log_{5} (6)$ by $\log_{6} (8)$ .

$\log_{2} (5) \log_{5} (6) \log_{6} (8)$

Step 2

Rewrite $\log_{2} (5)$ using the change of base formula.

Tap for more steps...

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 $0$ .

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

Step 2.2

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

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

Step 3

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

Tap for more steps...

Step 3.1

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 $0$ .

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

Step 3.2

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

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

Step 4

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

Tap for more steps...

Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

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

Step 6

Rewrite $\log_{6} (8)$ using the change of base formula.

Tap for more steps...

Step 6.1

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 $0$ .

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

Step 6.2

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

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

Step 7

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

Tap for more steps...

Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

Tap for more steps...

Step 11.1

Cancel the common factor.

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

Step 11.2

Divide $3$ by $1$ .

$3$