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
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? - - 0 4 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) - - - - 0 0 4 4 4. If < a < b. Sketch a graph that illustrates the difference between f(x) = a x and g(x) = b x
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 + 5 0. 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.788885. 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
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 7 + 4 = 40. 6 = ( x ) 4. 64 x = 6 5. 5 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.. 5 4 = 65. 5 = 4 4. 0 = 000 5. e 0.086 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 0000 5 5 = 8. ln = 0
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:
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 =
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) 8. 4 + log x 9. log x + log y If log 8 5 = 0. 774 and log 8 = 0. 58, determine the following: 0. log 8 5. log 8 45. log 8 0. log 8 5 4. log 8 5 8 5. log 8 0
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 4 60 7. 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) =
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. 5. 0 x = 50 6. 4log(x) 6 = 7. e 5 x = 4 8. 5ln(x ) = 5 9. 5 x+ = 4 x+ 0. 0 x = 5 x