Precalculus Examples

$\log_{4} (x) + \log_{16} (x - 1)$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} + \log_{16} (x - 1)$

Step 1.1.2

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

$\frac{\log (x)}{\log (4)} + \log_{16} (x - 1)$

Step 1.2

Rewrite $\log_{16} (x - 1)$ using the change of base formula.

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Step 1.2.1

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 $0$ .

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

Step 1.2.2

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

$\frac{\log (x)}{\log (4)} + \frac{\log (x - 1)}{\log (16)}$

Step 2

To write $\frac{\log (x)}{\log (4)}$ as a fraction with a common denominator, multiply by $\frac{\log (16)}{\log (16)}$ .

$\frac{\log (x)}{\log (4)} \cdot \frac{\log (16)}{\log (16)} + \frac{\log (x - 1)}{\log (16)}$

Step 3

To write $\frac{\log (x - 1)}{\log (16)}$ as a fraction with a common denominator, multiply by $\frac{\log (4)}{\log (4)}$ .

$\frac{\log (x)}{\log (4)} \cdot \frac{\log (16)}{\log (16)} + \frac{\log (x - 1)}{\log (16)} \cdot \frac{\log (4)}{\log (4)}$

Step 4

Write each expression with a common denominator of $\log (4) \log (16)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{\log (x)}{\log (4)}$ by $\frac{\log (16)}{\log (16)}$ .

$\frac{\log (x) \log (16)}{\log (4) \log (16)} + \frac{\log (x - 1)}{\log (16)} \cdot \frac{\log (4)}{\log (4)}$

Step 4.2

Multiply $\frac{\log (x - 1)}{\log (16)}$ by $\frac{\log (4)}{\log (4)}$ .

$\frac{\log (x) \log (16)}{\log (4) \log (16)} + \frac{\log (x - 1) \log (4)}{\log (16) \log (4)}$

Step 4.3

Reorder the factors of $\log (4) \log (16)$ .

$\frac{\log (x) \log (16)}{\log (16) \log (4)} + \frac{\log (x - 1) \log (4)}{\log (16) \log (4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{\log (x) \log (16) + \log (x - 1) \log (4)}{\log (16) \log (4)}$