Precalculus Examples

$\log_{5} (750 - \log_{5} (6))$

Step 1

Rewrite $\log_{5} (750 - \log_{5} (6))$ using the change of base formula.

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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 $b = 10$ .

$\frac{\log (750 - \log_{5} (6))}{\log (5)}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $\log_{5} (6)$ using the change of base formula.

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Step 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 $0$ .

$\frac{\log (750 - \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)})}{\log (5)}$

Step 2.1.2

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

$\frac{\log (750 - \frac{\log (6)}{\log (5)})}{\log (5)}$

Step 2.2

To write $750$ as a fraction with a common denominator, multiply by $\frac{\log (5)}{\log (5)}$ .

$\frac{\log (\frac{750 \log (5)}{\log (5)} - \frac{\log (6)}{\log (5)})}{\log (5)}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{\log (\frac{750 \log (5) - \log (6)}{\log (5)})}{\log (5)}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\frac{750 \log (5) - \log (6)}{\log (5)})}{\log (5)}$

Decimal Form:

$4.11235977 \dots$