Precalculus Examples

$\log_{x} (3) + \frac{\log (3)}{\log (\frac{x}{3})} + \frac{\log (3)}{\log (\frac{x}{81})}$

Step 1

To write $\log_{x} (3)$ as a fraction with a common denominator, multiply by $\frac{\log (\frac{x}{3})}{\log (\frac{x}{3})}$ .

$\frac{\log_{x} (3) \log (\frac{x}{3})}{\log (\frac{x}{3})} + \frac{\log (3)}{\log (\frac{x}{3})} + \frac{\log (3)}{\log (\frac{x}{81})}$

Step 2

Combine the numerators over the common denominator.

$\frac{\log_{x} (3) \log (\frac{x}{3}) + \log (3)}{\log (\frac{x}{3})} + \frac{\log (3)}{\log (\frac{x}{81})}$

Step 3

To write $\frac{\log_{x} (3) \log (\frac{x}{3}) + \log (3)}{\log (\frac{x}{3})}$ as a fraction with a common denominator, multiply by $\frac{\log (\frac{x}{81})}{\log (\frac{x}{81})}$ .

$\frac{\log_{x} (3) \log (\frac{x}{3}) + \log (3)}{\log (\frac{x}{3})} \cdot \frac{\log (\frac{x}{81})}{\log (\frac{x}{81})} + \frac{\log (3)}{\log (\frac{x}{81})}$

Step 4

To write $\frac{\log (3)}{\log (\frac{x}{81})}$ as a fraction with a common denominator, multiply by $\frac{\log (\frac{x}{3})}{\log (\frac{x}{3})}$ .

$\frac{\log_{x} (3) \log (\frac{x}{3}) + \log (3)}{\log (\frac{x}{3})} \cdot \frac{\log (\frac{x}{81})}{\log (\frac{x}{81})} + \frac{\log (3)}{\log (\frac{x}{81})} \cdot \frac{\log (\frac{x}{3})}{\log (\frac{x}{3})}$

Step 5

Write each expression with a common denominator of $\log (\frac{x}{3}) \log (\frac{x}{81})$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 5.1

Multiply $\frac{\log_{x} (3) \log (\frac{x}{3}) + \log (3)}{\log (\frac{x}{3})}$ by $\frac{\log (\frac{x}{81})}{\log (\frac{x}{81})}$ .

$\frac{(\log_{x} (3) \log (\frac{x}{3}) + \log (3)) \log (\frac{x}{81})}{\log (\frac{x}{3}) \log (\frac{x}{81})} + \frac{\log (3)}{\log (\frac{x}{81})} \cdot \frac{\log (\frac{x}{3})}{\log (\frac{x}{3})}$

Step 5.2

Multiply $\frac{\log (3)}{\log (\frac{x}{81})}$ by $\frac{\log (\frac{x}{3})}{\log (\frac{x}{3})}$ .

$\frac{(\log_{x} (3) \log (\frac{x}{3}) + \log (3)) \log (\frac{x}{81})}{\log (\frac{x}{3}) \log (\frac{x}{81})} + \frac{\log (3) \log (\frac{x}{3})}{\log (\frac{x}{81}) \log (\frac{x}{3})}$

Step 5.3

Reorder the factors of $\log (\frac{x}{81}) \log (\frac{x}{3})$ .

$\frac{(\log_{x} (3) \log (\frac{x}{3}) + \log (3)) \log (\frac{x}{81})}{\log (\frac{x}{3}) \log (\frac{x}{81})} + \frac{\log (3) \log (\frac{x}{3})}{\log (\frac{x}{3}) \log (\frac{x}{81})}$

Step 6

Combine the numerators over the common denominator.

$\frac{(\log_{x} (3) \log (\frac{x}{3}) + \log (3)) \log (\frac{x}{81}) + \log (3) \log (\frac{x}{3})}{\log (\frac{x}{3}) \log (\frac{x}{81})}$

Step 7

Apply the distributive property.

$\frac{\log_{x} (3) \log (\frac{x}{3}) \log (\frac{x}{81}) + \log (3) \log (\frac{x}{81}) + \log (3) \log (\frac{x}{3})}{\log (\frac{x}{3}) \log (\frac{x}{81})}$