Unit 1 - Number and Algebra
Get Ready
Questions
Evaluate the following two expressions:$$\log_{3}5=$$and$$\frac{\log 5}{\log 3}=$$What do you notice? What do you wonder?
Verify your thoughts with further examples.
Solutions
Notes
The change of base rule for logarithms is$$\log_{b}a=\frac{\log_{c}a}{\log_{c}b}$$In this way we can choose \(c\) to be any base we want, usually one that makes calculations easier.
Proof
Let$$m=\log_{b}a\implies b^{m}=a$$Therefore$$\begin{align}&\log_{c} b^{m}=\log_{c}a \\ &\implies m\log_{c}b=\log_{c}a \\ &\implies m=\frac{\log_{c}a}{\log_{c}b}\end{align}$$Q.E.D.
Examples and Your Turns
Example
Show that $$\log_{a}b=\frac{1}{\log_{b}a}$$
Your Turn
Show that $$\frac{1}{\log_{a}ab}+\frac{1}{\log_{b}ab}=1$$
Example
Evaluate $$\log_{3}5 \times\log_{5}3$$
Your Turn
Evaluate $$\log_{2}3 \times\log_{3}32$$
-
$$\begin{align}\log_{2}3 \times\log_{3}32 &=\log_{2}3 \times\frac{\log_{2}32}{\log_{2}3}\\&=\log_{2}32\\&=5\end{align}$$
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$$\begin{align}\log_{2}3 \times\log_{3}32 &= \frac{\log_{3}3}{\log_{3}2} \times \log_{3}32\\&=\frac{1}{\log_{3}2} \times \log_{3}32\\&=\frac{\log_{3}32}{\log_{3}2} \\&=\frac{\log_{3}2^{5}}{\log_{3}2} \\&=\frac{5\log_{3}2}{\log_{3}2} \\&=5\end{align}$$
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$$\begin{align}\log_{2}3 \times\log_{3}32 &=\frac{\log 3}{\log 2} \times\frac{\log 32}{\log 3}\\&=\frac{\log 32}{\log 2} \\&=\frac{\log 2^{5}}{\log 2} \\&=\frac{5\log 2}{\log 2} \\&=5\end{align}$$
Your Turn
Evaluate $$\log_{3}2 \times\log_{2}81$$
Your Turn
Evaluate $$\log_{6}10 \times\log 6$$
Your Turn
Evaluate $$\log_{125}8 \times\log_{5}8$$
Your Turn
Evaluate $$\frac{1}{\log_{2}6}+\frac{1}{\log_{3}6}$$
Your Turn
Evaluate $$\frac{1}{\log_{4}6}+\frac{1}{\log_{9}6}$$
Your Turn
Evaluate $$\log_{5}40-\frac{1}{\log_{8}5}$$
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.