Precalculus Examples

$\log_{2} (\frac{x^{4} + 32 x^{3} + 384 x^{2} + 2048 x + 4096}{(x - 17) (x - 5)})$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Make each term match the terms from the binomial theorem formula.

$\log_{2} (\frac{x^{4} + 4 \cdot 8 x^{3} + 6 \cdot 8^{2} x^{2} + 4 \cdot 8^{3} x + 8^{4}}{(x - 17) (x - 5)})$

Step 1.2

Factor $x^{4} + 4 \cdot 8 x^{3} + 6 \cdot 8^{2} x^{2} + 4 \cdot 8^{3} x + 8^{4}$ using the binomial theorem.

$\log_{2} (\frac{{(x + 8)}^{4}}{(x - 17) (x - 5)})$

Step 2

Rewrite $\log_{2} (\frac{{(x + 8)}^{4}}{(x - 17) (x - 5)})$ 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 $0$ .

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

Step 2.2

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

$\frac{\log (\frac{{(x + 8)}^{4}}{(x - 17) (x - 5)})}{\log (2)}$