6.3 Homework: properties of logarithms

In class, we demonstrated the first and third rules above. For instance, for the first rule:

${\text{log}}_{2}8={\text{log}}_2(2\times 2\times 2)=3$

${\text{log}}_{2}16={\text{log}}_2(2\times 2\times 2\times 2)=4$

${\text{log}}_2(8\times 16)={\text{log}}_2\left[\right(2\times 2\times 2\left)\right(2\times 2\times 2\times 2\left)\right]=7$

This demonstrates that when you multiply two numbers, their logs add .

Now, you come up with a similar demonstration of the second rule of logs, that shows why when you divide two numbers, their logs subtract .

${\text{log}}_{5}480=$

How can you use your calculator to test your answer to #2? (I’m assuming here that you can’t find ${\text{log}}_{5}480$ on your calculator, but you can do exponents.) Run the test—did it work?

${\text{log}}_5\left(\frac{2}{\text{15}}\right)$ $=$

${\text{log}}_5\left(\frac{\text{15}}{2}\right)=$

${\text{log}}_{5}64=$

${\text{log}}_5(5{)}^{23}=$

$5^{\left({\text{log}}_{5}23\right)}=$

${\text{log}}_a(x•x•x•x)$

${\text{log}}_a(x•1)$

${\text{log}}_a\left(\frac{1}{x}\right)$

${\text{log}}_a\left(\frac{x}{1}\right)$

${\text{log}}_a\left(\frac{x}{x}\right)$

${\text{log}}_a(x{)}^4$

${\text{log}}_a(x{)}^0$

${\text{log}}_a(x{)}^{-1}$

• a Draw a graph of $y={\text{log}}_{\frac{1}{2}}x$ . Plot at least 5 points to draw the graph.
• b What are the domain and range of the graph? What does that tell you about this function?