Precalculus Examples

$\frac{\log_{2} (8 a^{2} \cdot (16 b^{2}))}{\log_{3} (9 a^{2} \cdot (27 b^{3}))}$

Step 1

Multiply $16$ by $8$ .

$\frac{\log_{2} (128 a^{2} b^{2})}{\log_{3} (9 a^{2} \cdot 27 b^{3})}$

Step 2

Multiply $27$ by $9$ .

$\frac{\log_{2} (128 a^{2} b^{2})}{\log_{3} (243 a^{2} b^{3})}$

Step 3

Rewrite $\log_{2} (128 a^{2} b^{2})$ 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 $0$ .

$\frac{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}{\log_{3} (243 a^{2} b^{3})}$

Step 3.2

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

$\frac{\frac{\log (128 a^{2} b^{2})}{\log (2)}}{\log_{3} (243 a^{2} b^{3})}$

Step 4

Rewrite $\log_{3} (243 a^{2} b^{3})$ using the change of base formula.

Tap for more steps...

Step 4.1

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 $0$ .

$\frac{\frac{\log (128 a^{2} b^{2})}{\log (2)}}{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}$

Step 4.2

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

$\frac{\frac{\log (128 a^{2} b^{2})}{\log (2)}}{\frac{\log (243 a^{2} b^{3})}{\log (3)}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log (128 a^{2} b^{2})}{\log (2)} \cdot \frac{\log (3)}{\log (243 a^{2} b^{3})}$

Step 6

Combine.

$\frac{\log (128 a^{2} b^{2}) \log (3)}{\log (2) \log (243 a^{2} b^{3})}$