Precalculus Examples

$0.5 (\log_{4} (r - 7) - \log_{4} (r))$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$0.5 \log_{4} (\frac{r - 7}{r})$

Step 2

Simplify $0.5 \log_{4} (\frac{r - 7}{r})$ by moving $0.5$ inside the logarithm.

$\log_{4} ({(\frac{r - 7}{r})}^{0.5})$

Step 3

Apply the product rule to $\frac{r - 7}{r}$ .

$\log_{4} (\frac{{(r - 7)}^{0.5}}{r^{0.5}})$

Step 4

Rewrite $\log_{4} (\frac{{(r - 7)}^{0.5}}{r^{0.5}})$ using the change of base formula.

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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 $0$ .

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

Step 4.2

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

$\frac{\log (\frac{{(r - 7)}^{0.5}}{r^{0.5}})}{\log (4)}$