CHAPTER 4
4-3 logarithmic functions
objectives
Write equivalent forms for exponential and logarithmic functions.
Write, evaluate, and graph logarithmic functions.
Logarithmic functions
◦ How many times would you have to double $1 before you had $8? You could use an
exponential equation to model this situation. 1(2x) = 8. You may be able to solve
this equation by using mental math if you know 23 = 8. So you would have to
double the dollar 3 times to have $8.
◦ How many times would you have to double $1 before you had $512? You could
solve this problem if you could solve 2x = 8 by using an inverse operation that
undoes raising a base to an exponent equation to model this situation. This
operation is called finding the logarithm. A logarithm is the exponent to which a
specified base is raised to obtain a given value.
Logarithmic functions
You can write an exponential equation as a logarithmic
equation and vice versa.
Example #1
◦ Write each exponential equation in logarithmic form.
Exponential
Equation
35 = 243
25 = 5
104 = 10,000
6–1 =
ab = c
Logarithmic
Form
Example #2
◦ Write each exponential equation in logarithmic form.
Exponential
Equation
92= 81
33 = 27
0
x = 1(x ≠ 0)
Logarithmic Form
Student guided practice
◦ Do problems 2 to 5 in your book page 253
Example #3
◦ Write each logarithmic form in exponential equation.
Logarithmic
Form
log99 = 1
log2512 = 9
log82 =
log4
= –2
logb1 = 0
Exponential
Equation
Student guided practice
◦ Do problems 6 to 9 in your book page253
Logarithmic functions
◦ A logarithm with base 10 is called a common logarithm. If
no base is written for a logarithm, the base is assumed to be
10. For example, log 5 = log105.
Example #4
◦ Evaluate by using mental math.
log 0.01
Example #5
◦ Evaluate by using mental math.
log5 125
Example #6
◦ Evaluate by using mental math.
log5 1
5
Student guided practice
◦ Do problems 10 to13 in your book page 253
Properties of logarithms
◦ 1. loga (uv) = loga u + loga v
◦ 2. loga (u / v) = loga u - loga v
◦ 3. loga un = n loga u
◦ Many logarithmic expressions may be rewritten, either expanded or condensed, using the three
properties above. Expanding is breaking down a complicated expression into simpler
components. Condensing is the reverse of this process.
Properties of logs
◦ In less formal terms, the log rules might be expressed as:
◦ 1) Multiplication inside the log can be turned into addition outside the log, and vice
versa.
◦ 2) Division inside the log can be turned into subtraction outside the log, and vice versa.
◦ 3) An exponent on everything inside a log can be moved out front as a multiplier, and
vice versa.
Example
◦ Expand log3(2x).
Expand log4( 16/x ).
Student guided practice
◦ Do problems 1-6 in the worksheet
Condensing logs
◦ Simplify log2(x) + log2(y).
◦ Simplify log3(4) – log3(5).
◦ Simplify 3log2(x) – 4log2(x + 3) + log2(y).
Student guided practice
◦ Do odd problems from 13-20 in the worksheet
Graphs of log functions
◦ Because logarithms are the inverses of exponents, the inverse
of an exponential function, such as y = 2x, is a logarithmic
function, such as y = log2x.
The domain of y = 2x is all real
numbers (R), and the range is
{y|y > 0}. The domain of y =
log2x is {x|x > 0}, and the
range is all real numbers (R).
Example #7
◦ Use the x-values {–2, –1, 0, 1, 2}. Graph the function
and its inverse. Describe the domain and range of the
inverse function.
x
x
f(x) = 1.25
–2
–1
0
1
2
0.64
0.8
1
1.25
1.5625
Continue
◦ Inverse
x
0.64
0.8
1
f (x) = log1.25x
–2
–1
0
–1
1.25 1.5625
1
2
Example #8
◦ Use the x-values {–2, –1, 0, 1, 2}. Graph the function
and its inverse. Describe the domain and range of the
inverse function.
f(x) =
x
f(x) =(
)
x
1
2
x
–2
–1
0
4
2
1
1
2
Continue
◦ Inverse
Student guided practice
◦ Do problems 14 and 15 in your book page 253
Homework
◦ Do problems 17-25 in your book page 253
Closure
◦ Today we learned about logarithmic functions
◦ Next class we are going to have. A quiz and review
Have a great day