Skill 6 Exponential and Logarithmic Functions

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1 Skill 6 Exponential and Logarithmic Functions Skill 6a: Graphs of Exponential Functions Skill 6b: Solving Exponential Equations (not requiring logarithms) Skill 6c: Definition of Logarithms Skill 6d: Graphs of Logarithms Skill 6e: Properties of logarithms (product rule, quotient rule, power rule, change of base) Skill 6f: Logarithmic Equations (not requiring exponentials) Skill 6g: Exponential and Logarithmic Equations requiring inverse operations

2 Skill 6a: Graphs of Exponential Functions An exponential function is defined as an expression with a constant base with a variable exponent. The following are examples of exponential functions: f(x) = x g(x) = π x h(x) = x + In general an exponential function is of the form f(x) = a x, where a > 0 and a. Why is it necessary for a > 0? What can't a =?. Complete the table below for the function f(x) = x. Then graph the function at the right. x f(x) What is the domain of the basic exponential function? What is the range of the basic exponential function? What is the equation of the horizontal asymptote of the basic exponential function?. Complete the table for the function f(x) = 4 x.. Complete the table for f(x) = ( )x. Then graph the function above. Then graph the function above. x f(x) x f(x) If < a < b. Sketch a graph that illustrates the difference between f(x) = a x and g(x) = b x

3 5. If 0 < a < b <. Sketch a graph that illustrates the difference between f(x) = a x and g(x) = b x 6. If a >, how does the graph of f(x) = a x compare to the graphs of g(x) = ( a )x and h(x) = a x? Match the function below with the correct graph. 7. y = x 8. y = x 9. y = x y = ( )x. y = ( )x. y = ( )x. y = x 4 4. y = ( ) x 5. y = x 6. y = x 7. What is the domain, range, y-intercept, and the equation of the horizontal asymptote for f(x) = 4 x+. The number e is defined as the value of ( + n )n as n approaches infinity. e is an irrational number, but to ten decimal places it can be approximated as When e is the base of an exponential function, it is called the natural exponential function. 8. Sketch the graph of y = e 0.5x x - y - - 0

4 Skill 6b: Solving Exponential Equations (not requiring logarithms) Some exponential equations can be solved by rewriting constants values in terms of the base. Solve for x:. x = 8. 6 x = = ( x ) x = x = 5 6. x+ = 6 x 5 Skill 6c: Definition of Logarithms A logarithm is defined as the inverse of an exponential function.. f(x) = x, A) What is f()? B) What is f (8)? The exponential equation = 8 can be written as the logarithmic (or log) equation log 8 =. Rewrite the following exponential equations as logarithmic equations = = = e Note that log 0 x is usually written log x, so instead of writing log 0 00 =, write log 00 =. Also log e xis written ln x. Rewrite the following logarithmic equations as exponential equations. 5. log 7 = 6. log = 4 7. log = 8. ln = 0

5 Rewrite the following logarithmic equations as exponential equations and determine the value of x. 9. log 4 x = 0. log 4 64 = x. log x = 5. log 5 5 = x. log x 8 = 4. log,000,000 = x Skill 6d: Graphs of Logarithms Since a logarithm is the inverse of an exponential function, the graph of a y = log x is the reflection of the graph of y = x across the line y = x. x x x log x 0 0 For a basic logarithm: Domain: Range: Vertical Asymptote: X - Intercept: State the domain, range, x-intercept, and give the equation of the vertical asymptote for each function below:. f(x) = 5log (x). f(x) = log 5 (x 4) Domain: Range: Domain: Range: Vertical Asymptote: X - Intercept: Vertical Asymptote: X - Intercept:. f(x) = log (9x 7) 4. f(x) = ln( x + ) 4 Domain: Range: Domain: Range: Vertical Asymptote: X - Intercept: Vertical Asymptote: X - Intercept:

6 Match the function below with the correct graph. 5. y = log x A B C 6. y = log x 7. y = log ( x) 8. y = log x 9. y = log (x 4) D E F 0. y = log(x). y = log ( x 6 ). y = log ( x ) Match the function below with the correct graph.. y = ln x A B C 4. y = log 5 x 5. y = log x Skill 6e: Properties of Logarithms log(ab) = y, a = 0 m, and b = 0 n 0 y = ab = so, y = since a = 0 m and b = 0 n, m = and n =

7 So, log(ab) = y = m + n = log a + log b Product Rule of Logarithms log(ab) = log a + log b Also since log(a n ) = log(a a a) = log a + log a + log a = n log a Power Rule of Logarithms log(a n ) = n log a And recall log = log b b Quotient Rule of Logarithms log ( a ) = log a log b b Rewrite the following using the properties of logarithms:. log x. log x 00. log 4 x 0 4. log x y 5. log 5 ab x 6. log 7 7 Combine the following using the properties of logarithms into a single logarithm: 7. 4 log(x) + log(y) log (z) log x 9. log x + log y If log 8 5 = and log 8 = 0. 58, determine the following: 0. log 8 5. log log 8 0. log log log 8 0

8 Changing Bases: log a b = c can be rewritten as so, or, log a c = c log a = log b so, c = log b log a log a b = c = log b log a So with just a 'log' or 'ln' button on a calculator, any logarthin can be found. Change of Base Rule for Logarithms log a b = log b log a or log a b = ln b ln a Determine the following to four decimal places: 6. log log 8. log 7 ( 4) Skill 6f: Logarithmic Equations (not requiring inverse operations) Solve for x:. log(5) + log(x) = log() + log (0). log + log x = log 5 + log (x ). log (x 4) = log (5) log (x) 4. log x = log + log (x 4) 5. log (5 x) = 6. log (x + ) + log (x) =

9 Skill 6g: Logarithmic and Exponential Equations Exponential Functions and Logarithmic Functions are inverses of each other; f(x) = x g(x) = ln x f (x) = log x g (x) = e x Simplify the following expressions:. log (4x+). log 6 6 x. e ln(x 5) 4. ln e (9 4x) Solve each equation using inverse functions. Approximate solutions to decimal places when needed x = log(x) 6 = 7. e 5 x = ln(x ) = x+ = 4 x x = 5 x

Skill 6 Exponential and Logarithmic Functions

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