Math 3
Unit 7 Day 2 Notes – Logarithmic Functions
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Introduction to Logarithms
In its simplest form, a logarithm answers the question:
How many of one number do we multiply to get another number?
Example
How many 2s do we multiply to get 8?
Answer: 2 × 2 × 2 = 8, so we needed to multiply 3 of the 2s to get 8
So the logarithm is 3
How to Write it
We would write "the number of 2s you need to multiply to get 8 is 3" as
log2(8) = 3
So these two things are the same:
Try to remember the spiral relationship:
log2(8) = 3
Note:
*The base b is always a positive constant not equal to 1.
*Logs of negative numbers and zero are undefined since a positive number b raised to any power cannot
equal a negative number or 0.
Write in log form.
1.
2
5
25
2. 729
6
3
1
3.
2
3
1
8
4. 1 100
Evaluate.
1
32
5. log 4 64
6. log8 16
7. log 64
8. log9 27
9. log10 100
10. log3 81
A common logarithm is a log that uses base 10. Common logarithms are so common that they are often written
without the base notated:
log10 y
log y
While most scientific and graphing calculators have buttons for only the common logarithm and the natural
logarithm, other logarithms may be evaluated with the following change-of-base formula.
Change-of-base Formula:
Example : Evaluate log5 3. The change-of-base formula allows us to evaluate this expression using any other
logarithm, so we will solve this problem in two ways, using first the natural logarithm, then the common logarithm.
Natural Logarithm:
Common Logarithm:
Graphs of log functions
A log function is an inverse of an exponential function. Inverse functions are reflections of each other across the line
y=x.
11. Graph y
log 2 x
12. Graph y
log3 x