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6. 3 Logarithmic FunctionsObjectives: Write equivalent forms
for exponential and logarithmic equations.
Use the definitions of exponential and logarithmic functions to solve
equations.Standard: 2.8.11.S. Analyze
properties and relationships of functions.
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Logarithms are used to find unknown exponents in exponential models.
Logarithmic functions define many measurement scales in the sciences, including the pH, decibel, and Richter scales.
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With logarithms, you can write an exponential equation in an equivalent logarithmic form.
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For any positive base b, where b ≠ 1 if and only if x = log y
Ex 1.
a. Write in logarithmic form. ________________________
b. Write in exponential form. _______________________
c. Write 112 = 121 in logarithmic form. _________________________
d. Write log 6 36 = 2 in exponential form. _______________________
e. Write 7-2=1/49 in logarithmic form. __________________________
f. Write log 3 1/81= -4 in exponential form. _______________________
2 = log11 121
62 = 36
Log7 (1/49) = -2
3-4 = 1/81
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You can evaluate logarithms with a base of 10 by using the log key on a calculator.
Ex 2. Solve each equation for x. Round your answer to the nearest thousandth.
a). 10x= 1/109 b). x = log101/109 x = -2.037
c). 10x= 1.498 d). 10x= 7210 x = log10 1.498 x = log107210 x = .176 x = 3.858
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The inverse of the exponential function y = 10x is x = 10y.
To rewrite x = 10y in terms of y, use the equivalent logarithmic form, y = log 10 x.
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Examine the tables & graphs below to see the inverse relationship between y=10x and y = log10x.
y= 10x
y=x
y = log10x
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BeBelow summarizes the relationship between the domain and range of y = 10x and of y = log10 X.
• y = 10x
Domain: all Real #s
Range: all positive Real #s
• y = log10 X
Domain: all positive real #s
Range: all Real #s
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The logarithmic function y = log b x withbase b, or x = by, is the inverse of the exponential function y = bx, where b ≠ 1 and b > 0.
One-to-one Property of ExponentsIf bx = by, then x = y.
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Ex. 3 Find the value of v in each equation.
B.A.
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d. v = log464
4v = 64
4v = 43 (same base)
v = 3
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e. 2 = logv25 v2 = 25 v2=52
v = 5
f. 6 = log3v v = 36
v = 729
g. v = log10 1000 10v = 1000 10v = 103
v = 3
h. 2 = log7V V = 72
V = 49
I. 1 = log3v 31 = v 3 = v
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HomeworkPg. 374-375 #12-84 even