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CHAPTER 44-3 logarithmic functions
objectivesWrite 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
EquationLogarithmic
Form
35 = 243
25 = 5
104 = 10,000
6–1 =
ab = c
Example #2
◦Write each exponential equation in logarithmic form.
Exponential Equation
Logarithmic Form
92= 81
33 = 27
x0 = 1(x ≠ 0)
Student guided practice◦Do problems 2 to 5 in your book page 253
Example #3
◦Write each logarithmic form in exponential equation.Logarithmic
FormExponential
Equation
log99 = 1
log2512 = 9
log82 =
log4 = –2
logb1 = 0
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
15
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.
1f(x) = 1.25x
210–1–2x
0.64 0.8 1.25 1.5625
Continue◦ Inverse
210–1–2f–1(x) = log1.25x
1.56251.2510.80.64x
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 1 2
x –2 –1 0 1 2
f(x) =( ) x 4 2 1
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