Logarithmic Function

LOGARITHMIC FUNCTION

The Logarithmic Is the universe function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base B, must be raised, to produce the number x. 2

A special case of logarithmic functions is the natural logarithm, . It is defined as , and its derivative is , for . The base of the natural log is Euler's number e, such that .

“ Identifying Domain and Range of

LOGARITHMIC FUNCTION

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The domain of logarithmic function is real number greater than zero, and the range is real numbers”. “Remember that since the logarithmic functions is the inverse of the exponential function , the domain of logarithmic function is the range of exponential , and vise versa. In general, the function y=log b x where b, x >0 and b ≠ 1 is a continous and one to one function.” 5

y=a*logb(x-h)+k -In general log function the letter a tells us if the function has stretched, compressed, or reflected. -Also the letter b tells us the base of the log.

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f(x)= a logb (cx-h) +k

REMEMBER: The log is the inverse of an exponential. Domain of a log--------------Range of exp. Range of a log--------------Domain of exp. Recall: Range of a log function is always (-∞,∞)

For more complicated log functions, we need the argument of the log > 0. F(x) = a logb (cx-h) +k Range f = (-∞,∞) Domain f = {x│cx-h>0}

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EXAMPLE: g(x)= -log4 (x+2) +6 range g= (-∞,∞) domain g= {x│x > -2} = (-2,∞) SOLUTION: x+2>0 -2 -2 x >-2 EXAMPLE: h(x)= 4 log10 (3x-1) range h=(-∞,∞) domain h= {x│x > 1/3} = (1/3,∞)

SOLUTION: 3x-1>0 +1 +1 3x > 1 3 3 x>1 3

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X -Intercept & Y -Intercept 9

For all positive real numbers x and a , a ≠ 1 , there exists a real number y that such y= log x if and only if x=a

Thus,the logarithmic function with a base a , where a > 0 and a ≠ 1 , denoted by log , is defined by f(x)=y=log x if and only if a = x. Note that each other exponential function f(x)=a has an inverse function that is logarithmic function.

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1.y=2(x-1)(x+2)(x-3) f(x) y=2 (-1)(2)(-3)=12

(0,12) is on the graph

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Solve:log2 (2x+1)=3 Solution: log2 (2x+1) = 3 2x+1=2^3 2x+1=8 2x=7 x=7/2 12

Solve: 2 log4 , x = log4 , 9 Solution: Since both logarithms are to the base 4, 2 log4 x = log4 9 log4 x^2=log4 9 x^2=9 x=3 13

Express log5 3+2 log5 5 as a single logarithm.

Solution: log5 3+2 log5 5 = log5 3+ log5 5^2 = log5 3+ log5 25 = log5 75

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Solve: log2 (x+3) + log2 (x-3) = 4 Solution: log2 (x+3) + log2 (x-3) = 4 log2[(x+3)(x-3)] = 4 (x+3)(x-3) = 2^4 x^2 – 9 = 16 x^2 =25 x = √25 x= ± 5 15

Questions about logarithmic functions: Q: In the general log function y=a*logb(x-h)+k, what letter tells us the base of the log? a. a b. b c. h d. K Q: In the general log function y=a*logb(x-h)+k, what letter tells us if the function has stretched, compressed, or reflected? a. a

b. b c. h d. k

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Restine Joy Lumangyao Melvie May Go Trisha Mae Magante Alayssa Ashley Navales Lendy Mae Lanza 17