Precalculus Examples

$\frac{1}{1 + \log_{c} (a b)}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $\log_{c} (a b)$ 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 $0$ .

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

Step 1.1.2

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

$\frac{1}{1 + \frac{\log (a b)}{\log (c)}}$

Step 1.2

Write $1$ as a fraction with a common denominator.

$\frac{1}{\frac{\log (c)}{\log (c)} + \frac{\log (a b)}{\log (c)}}$

Step 1.3

Combine the numerators over the common denominator.

$\frac{1}{\frac{\log (c) + \log (a b)}{\log (c)}}$

Step 1.4

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

$\frac{1}{\frac{\log (c a b)}{\log (c)}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{\log (c)}{\log (c a b)}$

Step 3

Multiply $\frac{\log (c)}{\log (c a b)}$ by $1$ .

$\frac{\log (c)}{\log (c a b)}$