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83 Logarithmic Functions as Inverses 2011
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April 22, 2011
83 Logarithmic Functions as Inverses
83 Logarithmic Functions as Inverses
Objectives: Convert between exponential form and logarithmic form.Find the inverse of an exponential function.Apply logarithms to real world situations.
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Find the value of x in each example.1) 3x = 27 2) 5x = 625
3) 4x = 1024 4) 2x = 2048
We need an easier way to solve these problems, but how do you undo exponents?
Warmup:
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How do you solve this equation?
3x + 7 = 52
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Inverse functions are two functions whose operations undo each other.
f(x) = 3x 2 f 1(x) = x + 2 3
function inverse
Find f(f 1(4))
Find f 1(f(4))
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April 22, 2011
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bx = yFind the inverse of an exponential function.
switch the variables and solve for y
by = x
We run into the same problem, how do we solve for y?
We need an inverse function for exponents. This function is called a logarithm.
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pronounced "log base b of y equals x"
Logarithms are defined in the following way:
If bx = y, then logb y = x.
Using the definition rewrite the following exponential functions in logarithmic form.
25 = 32 32 = 9 43 = 64
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Using the definition, rewrite these logarithmic functions in exponential form.
log5 625 = 4 log10 10000 = 4 loge 54.598 ≈ 4
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Evaluating Logarithmic Functions
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Evaluate each logarithm.
a. log64 (1/32) b. log9 27 c. log10 100
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A COMMON LOGARITHM is a logarithm that uses base 10.
You can write the common logarithm log10 y as log y.
The log button on your calculator is used for base 10 calculations.
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Use your calculator to evaluate each logarithm to four decimal places.
a. log 9 b. log (3/7) c. log (10)
WHY does the third problem fail? Rewrite it in exponential form to make it easier to see.
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Finding Inverse Functions
How do we find the inverse of y = 3x?
Step 1: Write the exponential equation as a logarithm.
y = 3x is equal to log3 y = x
Step 2: Interchange x and y. (because we switch the domain and range to get the inverse)
log3 y = xlog3 x = y
Therefore, the inverse of y = 3x is y = log3 x.
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Try one!Find the inverse of y = (1/2)x.
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Logarithms and Exponential Functions are inverses. Therefore, we know that the graph of a logarithm is the graph of an exponential function reflected over the line y = x.
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Graph y = log2 x.
Step 1: Since y = 2x is the inverse of y = log2 x, start by graphing it.
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Graph y = log2 x.
Step 1: Since y = 2x is the inverse of y = log2 x, start by graphing it.
Step 2: Draw y = x.
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Graph y = log2 x.
Step 1: Since y = 2x is the inverse of y = log2 x, start by graphing it.
Step 2: Draw y = x.
Step 3: Choose a few points on y = 2x. Reverse the coordinates and plot the points of y = log2 x.
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We have graphed exponential functions using a ttable such as the ones below.
y = bx
101
x y
1b
1/b 101
x y
1b
1/b
It's inverse: y = logb xInverse
The points on the graph of y = logb x are:
(1/b, 1), (1, 0) and (b, 1).
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Logarithmic Properties
1) The Domain : { x | x > 0} Range : { y | y = (All Reals)}
2) There are no y intercepts and the x intercept is 1.
3) The y axis (x=0) is a vertical asymptote as x 0.
4) f(x) = logb x , 0 < b < 1, is an decreasing function and is onetoone.
5) The graph of f(x) contains the points (b,1), (1,0), (1/b,1).
0 < b < 1f (x) = logb x
(b,1)
(1,0)
(1/b,1)
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f (x) = logb x b > 1
(b,1)
(1,0)
(1/b,1)
Logarithmic Properties
1) The Domain : { x | x > 0} Range : { y | y = (All Reals)}
2) There are no y intercepts and the x intercept is 1.
3) The y axis (x=0) is a vertical asymptote as x 0.
4) f(x) = logb x , b > 1, is an increasing function and is onetoone.
5) The graph of f(x) contains the points (b,1), (1,0), (1/b,1).
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April 22, 2011
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homework8.1 8.3 Review
Worksheet8.1 - 8.3 Quiz
Monday!!!Time to complete the teacher eval!!!
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