Unit 1 - Number and Algebra
Required Prior Knowledge
Questions
Calculate
a) \(5^{-2} \)
b) \(\left(\frac{2}{3}\right)^{-3}\)
c) \(64^{\frac{1}{3}}\)
d) \(81^{\frac{3}{4}}\)
e) \(\left(\frac{27}{64}\right)^{-\frac{2}{3}}\)
Solutions
Get Ready
Questions
Solutions
Notes
Logarithms are the inverse of exponentials (also called indices and powers). Just like subtraction is the inverse of addition.
If we know that in exponential form we have $$y=b^{x}$$ then we can equivalently write this in logarithmic form as $$x=\log_{b}y$$
Note that the value of \(\log_{b}y\) is the power to which \(b\) must be raised to give \(y\).
From this basic definition, there are four characteristics which we can derive.
Try to complete the statements before revealing the answers.
a) \(\log_{b}1=\)
b) \(\log_{b}b=\)
c) \(\log_{b}b^{x}=\)
d) \(b^{\log_{b}x}=\)
-
a) \(\log_{b}1=0\)
b) \(\log_{b}b=1\)
c) \(\log_{b}b^{x}=x\)
d) \(b^{\log_{b}x}=x\)
-
a) This is asking what power of \(b\) gives a value of \(1\), which is \(0\).
b) This is asking what power of \(b\) gives a value of \(b\), which is \(1\).
c) This is asking what power of \(b\) gives a value of \(b\) to the power of \(x\), which is \(x\).
d) This one is a bit more subtle, and stems from the fact that the logarithm is the inverse of the exponential.
Examples and Your Turns
Example
Write \(2^{3}=8\) in logarithmic form.
Your Turn
Write \(7^{2}=49\) in logarithmic form.
Example
Write \(\log_{3}81=4\) in exponential form.
Your Turn
Write \(\log_{4}64=4\) in exponential form.
Your Turn
Solve each equation for \(x\).
$$x=\log_{16}4$$
$$\log_{b}1=x$$
$$\log_{7}x=2$$
$$\log_{x}32=\frac{5}{2}$$
Your Turn
Determine the value of each of these logarithms.
a) \(\log_{0.001}100\)
b) \(\log_{100}0.001\)
c) \(\log_{\frac{1}{2}}4\)
d) \(\log_{2\sqrt{7}}28\)
e) \(\log_{\sqrt{2}}32\)
f) \(\log_{256}32\)
g) \(\log_{0.3}0.027\)
Notes
For historical reasons, the logarithm base \(10\) was particularly important and used far more often than many other bases.
For this reason, a shorthand is used for the logarithm base \(10\). When a logarithm is written without a base stated explicitly, it is base \(10\).
That is $$\log x =\log_{10}x$$
Evaluate:
a) \(\log 100\)
b) \(\log 0.1\)
c) \(\log 0.0001\)
-
a) \(\log 100=2\)
b) \(\log 0.1=-1\)
c) \(\log 0.0001=-4\)
-
a) This is asking what power of \(10\) is \(100\).
b) This is asking what power of \(10\) is \(0.1=\frac{1}{10}\).
c) This is asking what power of \(10\) is \(0.0001=\frac{1}{10000}\).
Examples and Your Turns
Example
Approximate the value of \(\log 34\) (without a calculator), explaining your answer.
Your Turn
Explain why \(\log 500\) must be between \(2\) and \(3\).
-
We know that \(\log 100 = 2\) and \(\log 1000 = 3\) since \(10^{2}=100\) and \(10^{3}=1000\). Hence \(\log 500\) must be between these two values.
Using a calculator we can determine that \(\log 500 = 2.699\) to 3 decimal places.
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.
Taking it Deeper
Conceptual Questions to Consider
Why does \(\log_{b}1=0\) for all values of \(b\)?
Why can we not take the logarithm of a negative number?
If \(\log_{b}a=c\), explain what happens to \(c\) if:
\(b\) increases and \(a\) stays the same
\(a\) increases and \(b\) stays the same
\(b\) decreases and \(a\) stays the same
Common Mistakes / Misconceptions
Mixing up the base and the arguments in a logarithm.
Converting to exponential form incorrectly.
Forgetting that the base of a logarithm must be positive.
Connecting This to Other Skills
This skill is essential for later skills where we look at logarithms in more details (1.2, 1.10, 1.11 and 1.12). It is also really important for working with geometric sequences and series (1.5).
In later units, we will need logarithms when working with calculus in chapters 4 and 5 (4.11 and 5.7) as well as the more advanced differential equations, especially separable differential equations (8.6)
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?