Precalculus Examples

$\log_{5} (4 - \log_{5} (a + \log_{5} (b)))$

Step 1

Rewrite $\log_{5} (4 - \log_{5} (a + \log_{5} (b)))$ using the change of base formula.

Tap for more steps...

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 $0$ .

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

Step 1.2

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

$\frac{\log (4 - \log_{5} (a + \log_{5} (b)))}{\log (5)}$

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Rewrite $\log_{5} (a + \log_{5} (b))$ using the change of base formula.

Tap for more steps...

Step 2.1.1.1

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 $0$ .

$\frac{\log (4 - \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)})}{\log (5)}$

Step 2.1.1.2

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

$\frac{\log (4 - \frac{\log (a + \log_{5} (b))}{\log (5)})}{\log (5)}$

Step 2.1.2

Simplify the numerator.

Tap for more steps...

Step 2.1.2.1

Rewrite $\log_{5} (b)$ using the change of base formula.

Tap for more steps...

Step 2.1.2.1.1

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 $0$ .

$\frac{\log (4 - \frac{\log (a + \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)})}{\log (5)})}{\log (5)}$

Step 2.1.2.1.2

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

$\frac{\log (4 - \frac{\log (a + \frac{\log (b)}{\log (5)})}{\log (5)})}{\log (5)}$

Step 2.1.2.2

To write $a$ as a fraction with a common denominator, multiply by $\frac{\log (5)}{\log (5)}$ .

$\frac{\log (4 - \frac{\log (\frac{a \log (5)}{\log (5)} + \frac{\log (b)}{\log (5)})}{\log (5)})}{\log (5)}$

Step 2.1.2.3

Combine the numerators over the common denominator.

$\frac{\log (4 - \frac{\log (\frac{a \log (5) + \log (b)}{\log (5)})}{\log (5)})}{\log (5)}$

Step 2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{\log (5)}{\log (5)}$ .

$\frac{\log (\frac{4 \log (5)}{\log (5)} - \frac{\log (\frac{a \log (5) + \log (b)}{\log (5)})}{\log (5)})}{\log (5)}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{\log (\frac{4 \log (5) - \log (\frac{a \log (5) + \log (b)}{\log (5)})}{\log (5)})}{\log (5)}$

Step 2.4

Simplify the numerator.

Tap for more steps...

Step 2.4.1

Simplify $4 \log (5)$ by moving $4$ inside the logarithm.

$\frac{\log (\frac{\log (5^{4}) - \log (\frac{a \log (5) + \log (b)}{\log (5)})}{\log (5)})}{\log (5)}$

Step 2.4.2

Raise $5$ to the power of $4$ .

$\frac{\log (\frac{\log (625) - \log (\frac{a \log (5) + \log (b)}{\log (5)})}{\log (5)})}{\log (5)}$

Step 2.4.3

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

$\frac{\log (\frac{\log (\frac{625}{\frac{a \log (5) + \log (b)}{\log (5)}})}{\log (5)})}{\log (5)}$

Step 2.4.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log (\frac{\log (625 \frac{\log (5)}{a \log (5) + \log (b)})}{\log (5)})}{\log (5)}$

Step 2.4.5

Combine $625$ and $\frac{\log (5)}{a \log (5) + \log (b)}$ .

$\frac{\log (\frac{\log (\frac{625 \log (5)}{a \log (5) + \log (b)})}{\log (5)})}{\log (5)}$