Precalculus Examples

$\log_{4} (x - 2) + 4$

Step 1

Rewrite $\log_{4} (x - 2)$ 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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} + 4$

Step 1.2

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

$\frac{\log (x - 2)}{\log (4)} + 4$

Step 2

To write $4$ as a fraction with a common denominator, multiply by $\frac{\log (4)}{\log (4)}$ .

$\frac{\log (x - 2)}{\log (4)} + \frac{4 \log (4)}{\log (4)}$

Step 3

Combine the numerators over the common denominator.

$\frac{\log (x - 2) + 4 \log (4)}{\log (4)}$

Step 4

Simplify the numerator.

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Step 4.1

Simplify $4 \log (4)$ by moving $4$ inside the logarithm.

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

Step 4.2

Raise $4$ to the power of $4$ .

$\frac{\log (x - 2) + \log (256)}{\log (4)}$

Step 4.3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\frac{\log ((x - 2) \cdot 256)}{\log (4)}$

Step 4.4

Apply the distributive property.

$\frac{\log (x \cdot 256 - 2 \cdot 256)}{\log (4)}$

Step 4.5

Move $256$ to the left of $x$ .

$\frac{\log (256 \cdot x - 2 \cdot 256)}{\log (4)}$

Step 4.6

Multiply $- 2$ by $256$ .

$\frac{\log (256 x - 512)}{\log (4)}$