Mastery Check Solutions

Show What You Know

Name the rule or property used to complete each step.

Given: $\log_{b} \frac{\sqrt[3]{(x - y)}}{3 x}$

Explain

Step 1: $\log_{b} \frac{{(x - y)}^{\frac{1}{3}}}{3 x}$

Rewrite radicals as a fractional exponent.

Step 2: $\log_{b} {(x - y)}^{\frac{1}{3}} - (\log_{b} 3 x)$

Quotient rule

Step 3: $\frac{1}{3} \log_{b} (x - y) - (\log_{b} 3 x)$

Power rule

Step 4: $\frac{1}{3} \log_{b} (x - y) - (\log_{b} 3 + \log_{b} x)$

Product rule

Step 5: $\frac{1}{3} \log_{b} (x - y) - \log_{b} 3 - \log_{b} x$

Distributive property

Note

Remember to use your Formula Sheet. The first step does not have an official name and will need to be written out.

Write the expression as a single logarithm.
$\frac{1}{2} \log_{a} 5 + \frac{1}{2} \log_{a} x - \frac{1}{2} \log_{a} 7$

$\frac{1}{2} (\log_{a} 5 + \log_{a} x - \log_{a} 7) = \frac{1}{2} (\log_{a} 5 x - \log_{a} 7) \frac{1}{2} (\log_{a} \frac{5 x}{7}) = \log_{a} {(\frac{5 x}{7})}^{\frac{1}{2}} \log_{a} \sqrt{\frac{5 x}{7}}$

Note

Remember, the numerical arguments cannot be combined.

Solve: $2 \log_{3} x - \log_{3} 4 = \log_{3} 25$

$\begin{array}{rcl} \log_{3} x^{2} - \log_{3} 4 & = & \log_{3} 25 \\ \log_{3} \frac{x^{2}}{4} & = & \log_{3} 25 \\ \frac{x^{2}}{4} & = & 25 \\ x^{2} & = & 100 \\ x^{2} - 100 & = & 0 \\ (x + 10) (x - 10) & = & 0 \\ x & = & - 10, 10 \end{array}$

Note

Q: Explain why some values are excluded from the solution.
A: The argument must be greater than zero. If the argument is less than zero, that value must be excluded from the solution.

Say What You Know

In your own words, talk about what you have learned using the objectives for this lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Rewrite logarithmic expressions using the logarithmic properties.
Solve logarithmic equations.