Precalculus Examples

$\log_{4} (x) + \log_{0.25} (x - 1)$

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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} + \log_{0.25} (x - 1)$

Step 1.1.2

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

$\frac{\log (x)}{\log (4)} + \log_{0.25} (x - 1)$

Step 1.2

Rewrite $\log_{0.25} (x - 1)$ 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 $b = 10$ .

$\frac{\log (x)}{\log (4)} + \frac{\log (x - 1)}{\log (0.25)}$

Step 1.3

Log base $10$ of $0.25$ is approximately $- 0.60205999$ .

$\frac{\log (x)}{\log (4)} + \frac{\log (x - 1)}{- 0.60205999}$

Step 1.4

Move the negative in front of the fraction.

$\frac{\log (x)}{\log (4)} - \frac{\log (x - 1)}{0.60205999}$

Step 1.5

Multiply by $1$ .

$\frac{\log (x)}{\log (4)} - \frac{1 \log (x - 1)}{0.60205999}$

Step 1.6

Factor $0.60205999$ out of $0.60205999$ .

$\frac{\log (x)}{\log (4)} - \frac{1 \log (x - 1)}{0.60205999 (1)}$

Step 1.7

Separate fractions.

$\frac{\log (x)}{\log (4)} - (\frac{1}{0.60205999} \cdot \frac{\log (x - 1)}{1})$

Step 1.8

Divide $1$ by $0.60205999$ .

$\frac{\log (x)}{\log (4)} - (1.66096404 \frac{\log (x - 1)}{1})$

Step 1.9

Divide $\log (x - 1)$ by $1$ .

$\frac{\log (x)}{\log (4)} - (1.66096404 \log (x - 1))$

Step 1.10

Simplify $1.66096404 \log (x - 1)$ by moving $1.66096404$ inside the logarithm.

$\frac{\log (x)}{\log (4)} - \log ({(x - 1)}^{1.66096404})$

Step 2

To write $- \log ({(x - 1)}^{1.66096404})$ as a fraction with a common denominator, multiply by $\frac{\log (4)}{\log (4)}$ .

$\frac{\log (x)}{\log (4)} - \log ({(x - 1)}^{1.66096404}) \cdot \frac{\log (4)}{\log (4)}$

Step 3

Combine $- \log ({(x - 1)}^{1.66096404})$ and $\frac{\log (4)}{\log (4)}$ .

$\frac{\log (x)}{\log (4)} + \frac{- \log ({(x - 1)}^{1.66096404}) \log (4)}{\log (4)}$

Step 4

Combine the numerators over the common denominator.

$\frac{\log (x) - \log ({(x - 1)}^{1.66096404}) \log (4)}{\log (4)}$