Target Goals: 1. Use the relationship between exponential and logarithmic functions to change them...
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![Page 1: Target Goals: 1. Use the relationship between exponential and logarithmic functions to change them into the opposite form 2. Evaluate logarithms with and.](https://reader030.fdocuments.us/reader030/viewer/2022020801/5697c0081a28abf838cc6e59/html5/thumbnails/1.jpg)
4.3 – LOGARITHMIC FUNCTIONSTarget Goals:
1. Use the relationship between exponential and logarithmic functions to change them into the opposite form
2. Evaluate logarithms with and without a calculator.
3. Solve logarithmic equations
![Page 2: Target Goals: 1. Use the relationship between exponential and logarithmic functions to change them into the opposite form 2. Evaluate logarithms with and.](https://reader030.fdocuments.us/reader030/viewer/2022020801/5697c0081a28abf838cc6e59/html5/thumbnails/2.jpg)
EXPONENTIAL VS. LOGARITHMIC FORM
Exponential Form Logarithmic Form
ya x logay x
0, 1a a
![Page 3: Target Goals: 1. Use the relationship between exponential and logarithmic functions to change them into the opposite form 2. Evaluate logarithms with and.](https://reader030.fdocuments.us/reader030/viewer/2022020801/5697c0081a28abf838cc6e59/html5/thumbnails/3.jpg)
EX 1) CHANGE EACH EXPONENTIAL EXPRESSION TO AN EQUIVALENT EXPRESSION INVOLVING A LOGARITHM:
64.5 x 9qe 5 14s
4.5log 6x log 9e q log 14 5s
ln 9 q
![Page 4: Target Goals: 1. Use the relationship between exponential and logarithmic functions to change them into the opposite form 2. Evaluate logarithms with and.](https://reader030.fdocuments.us/reader030/viewer/2022020801/5697c0081a28abf838cc6e59/html5/thumbnails/4.jpg)
EX 2) CHANGE EACH LOGARITHMIC EXPRESSION TO AN EQUIVALENT EXPRESSION INVOLVING AN EXPONENT:
log 6 5x ln 2b 5log 4 x
5 6x log 2e b 5 4x
2e b
![Page 5: Target Goals: 1. Use the relationship between exponential and logarithmic functions to change them into the opposite form 2. Evaluate logarithms with and.](https://reader030.fdocuments.us/reader030/viewer/2022020801/5697c0081a28abf838cc6e59/html5/thumbnails/5.jpg)
EX 3) FIND THE EXACT VALUE OF EACH:
3log 81 2
1log
32
3 81x
x x
43 3x 4x
12
32x
12 32x 152 2x
5x
![Page 6: Target Goals: 1. Use the relationship between exponential and logarithmic functions to change them into the opposite form 2. Evaluate logarithms with and.](https://reader030.fdocuments.us/reader030/viewer/2022020801/5697c0081a28abf838cc6e59/html5/thumbnails/6.jpg)
DOMAIN AND RANGE OF LOG FUNCTIONS! Domain of a logarithmic function = range of the exponential function =
(0,∞)
Range of a logarithmic function = domain of the exponential function = (-∞, ∞)
Ex 4) Find the domain of each logarithmic function:3( ) log ( 8)f x x 5
2( ) log
3
xf x
x
8 0x
8x
(8, )
20
3
x
x
2 0x 2x
(2,3)
![Page 7: Target Goals: 1. Use the relationship between exponential and logarithmic functions to change them into the opposite form 2. Evaluate logarithms with and.](https://reader030.fdocuments.us/reader030/viewer/2022020801/5697c0081a28abf838cc6e59/html5/thumbnails/7.jpg)
PROPERTIES OF A LOGARITHMIC FUNCTION:
1. The domain is the set of positive real numbers; the range is all real numbers.
2. The x-intercept of the graph is 1. There is no y-intercept.
3. The y-axis (x = 0) is a vertical asymptote of the graph.
4. A logarithmic function is decreasing if 0 < a < 1 and increasing if a > 1.
5. The graph of f contains the points (1, 0), (a, 1) and (1/a, -1).
6. The graph is smooth and continuous, with no corners or gaps or holes.
( ) logaf x x
![Page 8: Target Goals: 1. Use the relationship between exponential and logarithmic functions to change them into the opposite form 2. Evaluate logarithms with and.](https://reader030.fdocuments.us/reader030/viewer/2022020801/5697c0081a28abf838cc6e59/html5/thumbnails/8.jpg)
EX 5) SOLVE THE FOLLOWING EQUATIONS:3log (4 7) 2x log 64 2x 2 5xe
23 4 7x 9 4 7x 16 4x
4x
2 64x 8x 8x
ln 5 2x
ln 5
2x