Example. Determine the inverse of the given function (if it exists). f(x) = 3

Transcription

1 Example. Determine the inverse of the given function (if it exists). f(x) = g(x) = p x + x We know want to look at two di erent types of functions, called logarithmic functions and exponential functions. It will turn out that these two types of functions are inverses of each other, i.e. the inverse of a logarithmic function will be an exponential function and the inverse of an exponential function will be a logarithmic function. Definition. The function Exponential Functions (Section.) x is a real number, a > 0, and a =,iscalledtheexponential function with base a. Example. Consider the exponential functions Find the following output values. f(x) = x and g(x) = x and h(x) =e x. f( ) = g( ) = h( ) = f(0) = g(0) = h(0) = f() = g() = h() =

2 Let s draw the graphs of f(x) = x and f(x) = x : Now, let s draw the graphs of f(x) = x and f(x) = So graphs of exponential functions look like: --- x :

3 What does log a x mean???? Logarithms (Section.) Example. Calculate the following. log log log log log log Definition. The function is the logarithmic function having base a, wherea>0isarealnumber. Two bases that occur often have special notations:

4 0 Example. Compute the following. Round your answers to two decimal places. log ln log 0 ln 0 Your calculator has buttons for ln x and log x, buthowdoyoucalculateotherbaselogarithms? Example. Compute the following. Round your answers to two decimal places. log log log 0 You can convert between logarithmic and exponential equations using the following:

5 Example. Convert the following logarithmic equations to exponential equations. log = log=0. ln 0 = x Example. Convert the following exponential equations to logarithmic equations. e = t = 0 =000 Graphs of Inverse Functions (Section.) Let s compare the graphs of two inverse functions. Graph f(x) =x andf (x) = x + on the same coordinate axes:

6 Let s try another pair of inverse functions. Graph f(x) =x +andf (x) = p x same coordinate axes: onthe Since f(x) =a x and f (x) =log a (x), we can use the graph of f(x) =a x to graph f (x) = log a (x). Example. Sketch the graph of each logarithmic equation. y =log x y =log x

7 Application to Compound Interest (Section.) If P dollars are invested at an interest rate r (written as a decimal), compounded n times per year, then the amount A of money in the account after t years is given by: A = P + r nt n Example. If $00 is invested at a rate of.%, compounded monthly, how much money will be in the account after years? Example. If $00 is invested at a rate of.%, compounded semiannually, how much money will be in the account after years? As the number of times you compound interest per year increases, the amount in the account after t years also increases. Consider investing $00 at a rate of % for a time period of years. So A = n n

8 If interest is compounded continuously, then the amount A in the account after t years is: A = Pe rt Example. If $00 is invested at a rate of.%, compounded continuously, how much money will be in the account after years? Example. If $00 is invested at a rate of.%, compoundedcontinuously, how much money will be in the account after years? Solving Exponential and Logarithmic Equations (Section.) Logarithm and exponential functions are inverses, so: log a (a x )=x and a log a (x) = x Example. Solve the following equations: x+ = log (x ) =

9 Example. Solve the following equations: x = log ( x) = x =0 lnx = Properties of logarithms: log a M +log a N =log a MN log a M log a N =log a M N p log a M =log a M p

10 So, how do we solve logarithmic equations like log (x +) log x = Example. Solve the following equations: log (x +)+log (x ) = log x +log(x +)=log

11 One more example... solve ln(x + ) + ln(x ) = ln x Let s take a look back at solving exponential equations. Example. Solve the following equations: 000e 0.0t =000 0 (.) x =0

12 Example. Suppose that $,00 is invested at an interest rate of.%, compounded quarterly. How long would it take for the amount of money in the account to become $,000? Example. Suppose that $,000 is invested at an interest rate of.%, compounded continuously. How long would it take for the amount of money to double?