Download log x b y x = ⇔ =

MA 15800
Lesson 15 Notes
Logarithmic Functions
Summer 2016
Every 1-1 function (where each x yields 1 and only 1 y and each y is the result of one and only 1 x) will
have an inverse function. Since an exponential function is 1-1, it has an inverse. This inverse is a
logarithmic function.
Basic Exponential function
Inverse: Basic Logarithmic function
𝒚 = 𝒃𝒙
𝒙 = 𝒃𝒚
Because the functions above look so similar, mathematicians developed a new notation for logarithmic
functions.
y
b
The form of the logarithmic function on the left is in exponential form and the form on the right is in
logarithmic form.
x  b  y  log x
Definition of a Logarithmic Function with base b:
For all 𝑥 > 0 and 𝑏 > 0 (𝑏 ≠ 1), the following is called a logarithmic function and is the inverse of an
exponential function.
The value of y is either form is called a logarithm.
y  logb x is equivalent to b y  x
Ex 1: (a)
4
Convert each exponential form to logarithmic form.
5
3  81
m  (n  1)

(2a) y  7  p 2
3
4
2

9
16
ar  9
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MA 15800
(b)
Lesson 15 Notes
Logarithmic Functions
Summer 2016
Convert each logarithmic form to exponential form.
log 2 16  4
log1/2
 3
1
8
logb m  5  x
log r (a  2)  n
log q (2mn)  12
log x  2 200  rs
Ex 2: The number N of bacteria in a certain culture after x hours is given by 𝑁 = (1000)2𝑥 . Express x
as a logarithmic function with base 2.
As seen in the top box on page 1 (𝒙 = 𝒃𝒚 ), a logarithm is an exponent!! When asked to find a
logarithm, you are asked to find the appropriate exponent on the base that equals the number given.
Ex 3: Find each logarithm, if possible. I not possible, state the logarithm does not exist.
log10 10, 000 
log 4 64 
log9 3 
log1/2 16 
log12 1 
log3 (2) 
log 2
 
1
32
log8 4096 
Properties of logarithms (part 1):
log a x
Reason
Illustration
(1)
log 𝑎 1 = 0
𝑎0 = 1
log 4 1 = 0
(2)
log 𝑎 𝑎 = 1
𝑎1 = 𝑎
log 7 7 = 1
(3)
log 𝑎 𝑎 𝑥 = 𝑥
𝑎𝑥 = 𝑎𝑥
log 2 28 = 8
𝑎log𝑎 𝑥 = 𝑥
If 𝑦 = log 𝑎 𝑥, then 𝑥 = 𝑎 𝑦 .
Make a substitution…
𝑥 = 𝑎log𝑎 𝑥
5log5 2 = 2
Property of
(4)
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MA 15800
Lesson 15 Notes
Logarithmic Functions
Summer 2016
Ex 4: Simplify the following. If it does not exist, write DNE.
log5 1 
log12 12 
log ( 4) 1 
log10 103 
4log4 9 
log10 (100) 
Since the logarithmic function is the inverse of the exponential function, they have symmetry about the
line with equation 𝑦 = 𝑥. Examine the following graphs.
or 𝑦 = log 2 𝑥
Below are the same two graphs (the exponential function and its inverse logarithmic function) on the
same coordinate system. It is easy to see the symmetry about the line 𝑦 = 𝑥.
(continued on the next page)
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MA 15800
Lesson 15 Notes
Logarithmic Functions
Summer 2016
Characteristics of a Logarithmic function’s graph:
(1)
The x-intercept is 1.
(2)
The graph is increasing if 𝑏 > 1, decreasing if 0 < 𝑏 < 1.
(3)
The domain is all positive numbers, 𝐷 = (0, ∞), so the graph is to the right of the y-axis.
(4)
The y-axis ia a vertical asymptote.
You will notice that your TI-30XA calculator has two keys to find logarithms. One is LOG; this finds
logarithms of base 10. The other is LN. This finds logarithms of the irrational number e. A logarithm
of base 10 is called a common logarithm. A logarithm of base e is called a natural logarithm
There is special notation for common logarithms and natural logarithms.
log10 x  log x for every x  0
loge x  ln x for every x  0
The LOG key on your TI-30XA finds common logarithms, the exponent on 10 that would equal the
number entered.
The LN key on your TI-30XA find natural logarithms, the exponent on e that would equal the number
entered.
Ex 5: Use your TI-30XA calculator to approximate each to 4 decimal places.
log 75 
log 0.125 
ln 22.8 
ln
 
2
3
Our TI-30Xa calculator has keys for base 10 and base e. However, there are times that other bases are
reasonable when evaluating some logarithmic expressions. We can use a change of base formula to
convert any logarithm to another expression using a different base. See the theorem below.
Theorem: Change of Base Formula
If 𝑢 > 0 and if a and b are positive real numbers other than 1, then
logb u 
log a u
log a b
Caution: Do not confuse the change of base formula with the quotient rule of logarithms. See the
next comparison.
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MA 15800
Lesson 15 Notes
Logarithmic Functions
Summer 2016
m
log b    log b m  log b n
n
log a m
 m  log a m
log b   
, log b (m  n) 
log a n
 n  log a n
Since our TI-30XA calculator has keys for common logarithms (base 10) and natural logarithms (base
e), these are the bases you will most often use with the change of base formula.
log u
ln u
logb u 
or logb u 
log b
ln b
However, be aware that another base may be more reasonable. Examine this example.
log9 243  ?
9 and 243 can both be written as powers of 3, so I will use base 3 with the change of base formula.
log3 243
log 9 243 
(35  243 and 32  9)
log3 9

5
2
 1 
Ex 6: Find the exact value of log 25 
 by using the change of base formula with base 5.
 125 
Ex 7: Use the change of base formula and your TI-30XA calculator to approximate each to 4 decimal
places.
(a) log3 7 
(b) log13 20 
(c) log0.5 3.93 
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MA 15800
Lesson 15 Notes
Logarithmic Functions
Summer 2016
Ex 8: The function 𝑓(𝑥) = 29 + 48.8log(𝑥 + 1) is a model for the percentage of adult height attained
by a boy who is x years old. Approximately what percentage of adult height (to the nearest 10th of a
percentage) has a boy of age 11 achieved? (Notice that this model is using common logiarhtms.)
Ex 9: The function T ( x)  13.4ln x 11.6 models the temperature increase in degrees Fahrenheit after
x minutes in an enclosed vehicle when the outside temperature is from 72° to 96°. Use the function to
approximate the temperature increase after 45 minutes, if the outside temperature is 85°. Round to the
nearest tenth of a degree.
The Richter scale is used to measure the intensity of an earthquake. The Richter scale is based on
common logarithms. An earthquake of 5.0 is 10 times the magnitude of an earthquake of 4.0. The
Richter scale is defined as below, where the magnitude R of an earthquake is a common log of the ratio
of the intensity I to the intensity of a very, very tiny magnitude called 𝐼0 .
I
R  log
I0
Ex 10: Below are some given intensities of a few earthquakes. What are the measurements of the
quakes on the Richter scale?
1000I 0
475I 0
5876I 0
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MA 15800
Lesson 15 Notes
Logarithmic Functions
Summer 2016
Solving Logarithmic Equations: There are two types of logarithmic equations.
Type 1: logb m  logb n  m  n
As seen above, if the equation can be written with a logarithm of the same base on each side of the
equation, then the arguments are equal.
Type 2: logb m  r  br  m
As seen above, if the equation has a logarithm on one side and a number on the other side, convert to
exponential form and then solve.
Ex 11:
Solve each equation.
(a)
log( x  2)  log( x2  10)
(b)
Remember to always check possible solutions to
logarithmic equations. The checks must yield
positive arguments. Disregard any ‘solution’ that
leads to a non-positive argument.
log7 ( x  5)  log7 (6 x)
Ex 12:
Solve each equation.
log3 ( x  4)  4
(a)
(b)
log 2 (5  x)  4
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MA 15800
Lesson 15 Notes
Logarithmic Functions
Summer 2016
Ex 13: A logarithmic function (or its graph) contains the ordered pairs (8,3). Find an equation for the
function of the form 𝑓(𝑥) = log 𝑎 𝑥. (In other words, determine the base a.)
Ex 14: Sketch a graph of this logarithmic function. Hint: Convert to exponential form.
𝑦 = log 3 𝑥
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MA 15800
Lesson 15 Notes
Logarithmic Functions
Summer 2016
Ex 15: Approximate x to three significant figures.
a) log x  3.6274
b) ln x  1.6
c) log x  0.95
d ) ln x  3.73
Ex 16: The population 𝑁(𝑡) (in millions) of the United States t years after 1980 may be approximated
by the formula N (t )  231e0.0103t . During what year will be population be twice what it was in 1980?
Ex 17: If interest is compounded continuously at the rate of 6% per year, approximate the number of
years it will take an initial deposit of $6000 to grow to $25,000.
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