Explore: Contract Expressions with the Properties of Logs Solutions

Contract Expressions with the Properties of Logs Solutions

The goal for contracting a logarithmic expression is to combine all logs of the same base into a single logarithm .  
Work backward from multiple terms to one term using the properties of logarithms. 
It is recommended that you contract logs using the properties in this order: 
- Power Rule 
- Product Rule 
- Quotient Rule  
Instructions for contracting expressions might say “ contract ” or “ rewrite as a single log .”

Note

There are exceptions to this order, but in general, this is the order used in this lesson.

In this lesson, all logs have the same base.

Example 3

Rewrite as a single log.

$\frac{3}{4} \log_{b} x + \frac{1}{4} \log_{b} y$

Implement

$\log_{b} x^{\frac{3}{4}} + \log_{b} y^{\frac{1}{4}} \log_{b} x^{\frac{3}{4}} y^{\frac{1}{4}} = \log_{b} {(x^{3} y^{1})}^{\frac{1}{4}} \log_{b} \sqrt[4]{x^{3} y}$

Explain

Power Rule 
Product Rule 
Fractional exponent to a radical

$\log_{b} 5 + 2 \log_{b} x - \log_{b} y - 4 \log_{b} z$

Implement

$\log_{b} 5 + \log_{b} x^{2} - \log_{b} y - \log_{b} z^{4} \log_{b} 5 x^{2} - (\log_{b} y + \log_{b} z^{4}) \log_{b} 5 x^{2} - \log_{b} y z^{4} \log_{b} \frac{5 x^{2}}{y z^{4}}$

Explain

Power Rule 
Product Rule 
Quotient Rule

Example 4

Contract.

$2 \log_{9} (x + 4) + 3 \log_{9} (x - 2) - 4 \log_{9} x$

$\log_{9} {(x + 4)}^{2} + \log_{9} {(x - 2)}^{3} - \log_{9} x^{4} \log_{9} {(x + 4)}^{2} {(x - 2)}^{3} - \log_{9} x^{4} \log_{9} \frac{{(x + 4)}^{2} {(x - 2)}^{3}}{x^{4}}$