Precalculus Examples

$\frac{\ln (3)}{\ln (5)} - 7$

Step 1

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{\ln (5)}{\ln (5)}$ .

$\frac{\ln (3)}{\ln (5)} - 7 \cdot \frac{\ln (5)}{\ln (5)}$

Step 2

Combine fractions.

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Step 2.1

Combine $- 7$ and $\frac{\ln (5)}{\ln (5)}$ .

$\frac{\ln (3)}{\ln (5)} + \frac{- 7 \ln (5)}{\ln (5)}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{\ln (3) - 7 \ln (5)}{\ln (5)}$

Step 3

Simplify the numerator.

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Step 3.1

Simplify $- 7 \ln (5)$ by moving $7$ inside the logarithm.

$\frac{\ln (3) - \ln (5^{7})}{\ln (5)}$

Step 3.2

Raise $5$ to the power of $7$ .

$\frac{\ln (3) - \ln (78125)}{\ln (5)}$

Step 3.3

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

$\frac{\ln (\frac{3}{78125})}{\ln (5)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (\frac{3}{78125})}{\ln (5)}$

Decimal Form:

$- 6.31739380 \dots$