TBF General Mathematics - I Lecture – 5 : Exponential and Logarithmic Functions Prof. Dr. Halil İbrahim Karakaş Başkent University

Exponential Functions. Let b  ℝ, b > 0, b  1. The function f defined by the equation İs called the exponential function with base b. The domain of the exponential function, with any base, is ℝ, and the range is (0,  ).). x y (0,0) 1 1 x y 1 1 x y x y

x y 1 1 x y Some Properties of Exponential Functions. y-intercept is (0, 1). x-axis (y=0) is horizontal asymptote. If b > 1, then bx bx increases as x increases ; If 0 < b < 1, then bx bx decreases as x increases. b x b y = b x+y, bx bx = by by  x = y. If x  0, ax ax = bx bx  a = b. Continuous on (- ,  ).

The graph of Consider the following fundamental transformation: 2 units horizontal shift to the right x y (0,0) 1 x y (2,1) Note that the same graph could be obtained by shrinking the graph of y = 2x2x

The graph of x y (0,0) 1 x y (1,3) 2 1 unit vertical shift upwards The graph of or x y (0,0) (1,0) 1 x y (0,0) (1,1)

The graph of over the interval [-1,1]. x y (0,0) (1,0) 1 x y 1 (0,0) (-1,3/4)

The number e The exponential function with base e is called the natural exponential function. x y (0,0) 1 It can be shown that exists. It is easy to seefor x > 1.

Interest. The fee paid to use another’s money. The interest is usually computed as a percent (called the interest rate) of the amount of money used (called the principal or present value) over a given period of time. The interest rate is usually considered annual and expressed as decimal. If the principal is P, interest rate is r, then at the end of t years the interest I is given by simple interest The amount A in the account at the end of t years (called the future value is given by simple interest Example. If 1000 TL is invested into an account paying 10% annual interest rate, the amount in the account at the end of 3 years will be

Compound Interest. If at the end of each payment period, the interest is reinvested at the same rate, then the interest earned as well as the principal will earn interest during the next payment period. This kind of inerest is called the compound interest. Let us assume that a principal P is invested at an annual rate r and compounded m times a year, at the end of the first compounding period the amount in the account reaches to At the end of the second compounding period the amount in the account reaches to Continuing in this way, the amount in the account at the end of one year (m (m periods)will be

At the end of two years And at the end of t years Compound interest Example. If TL is invested into an account paying 10 % compounded monthly, at the end of 3 years the amount in the account will be P= 1000, r=0.1, m = 3 and t = 3.3. TL. m : number of compounding periods. approximate

In the compound interest formula, : rate per compounding period, mt = n : toal number of compounding periods Then the compound interest formula becomes Example. If TL is invested into an account paying 10 % compounded every six months (semi annually), at the end of 3 years the amount in the account will be TL. P = 1000, m = 2, i = (0.1)/2 = So n= mt = 2. 3 = 6.6. Example. If TL is invested in an account paying 10 % compounded every three months, at the end of 3 years the amount in the account will be TL. A= 1000, r= 0.1, m = 4, t = 3.3. set

In compound interest formula keep P, r and t fixed and consider the limit as m  .. As m  ,, as well. So we have This computation leads to the formula A = Pe rt which is known as the continuous compound interest formula.

ContiniousCompound Interest. If a principal P is barrowed at an annual rate of r and if interest is compounded continuously, then after t years the barrower will owe the lender an amount given by P : principal r : interest rate (written as a decimal) A : final amount t : time Example. If TL is invested at 10 % compounded continuously, what amount will be in the account after 3 years? TL.

Simple InterestCompound InterestContinuous Compound Interest

Logarithmic Functions. Let us recall that the domain of the exponential function (with base b)b) is ℝ and its range is (0,  ). We can also note that each y  (0,  ) is the image of one and only one number x  ℝ. In other words, for each y  (0,  ) there is a unique x  ℝ such that y = bx bx. Given y  (0,  ) the number x  ℝ for which y = b x is called the logarithm of y to the base b. Then we write Thus The function defined by the equation is called the logarithmic function with base b.b. The domain of this function is (0,  ), and the range is ℝ.ℝ.

The definition of the function log b İs the fundamental relation which expresses the connection between exponential and logarithmic functions with base b. Every computation about logarithmic functions can be achived by using this relation. For example, if b > 0 and b  1, then log b 1 = 0. For

Recall the definition of the logarithmic function with base b:b: We see from this definition that, if (u, v) is a point on the graph of the logarithmic function with base b, then the point (v, u) u) is on the graph of the exponential function with base b; and conversely, If (u, v) v) is a point on the graph of the exponential function with base b, then the point (v, u) u) is on the graph of the logarithmic function with base b.b. On the other hand, the points (u, v)v) and (v, u) are symmetric about the line y = x. Therefore, the graphs of the exponential and logarithmic functions with base b are symmetric about the line y = x.x. x y (0,0) (u, v) (v,u) y = x

x y (0,0) 1 1 x y 1 1 y = x

2 To sketch graphs inwolwing logarithmic or exponential functios, we may get use of fundamental transformations. Example. Let us sketch y=log 2 x  (x-1) 1 x y (0,0)

Example. 1 x y (0,0) (1,3) y = log 2 x  y = 3 + x If we want to consider the graph of this function over an interval, for instance over the interval [1,2], we sketch that part of the graph lying above that interval; for our example, the part for which 1 ≤ x ≤ 2.2. The graph over the interval [1,2] is traced as a thick blue curve. 2

Example. y=log 2 x  (x-1)  y=3+log 2 (x-1) 1 x y (0,0) (0,3)

Properties of the Logarithmic Function.  x-intercept is (1,0).  If b >1, then log b x increases as x increases.  If b >1, then log b x  -  as x  0 + and log b x   as x  .  If 0 < b < 1, then log b x   as x  0 + and log b x  -  as x  .  If 0 < b < 1, then log b x decreases as x increases.

All the properties listed in the previous slide can be seen directly from the definitions. The proof of can be given as The proof of : As for the proof of the property concerning the logarithm of the product of two numbers: Example. Logarithmic calculations

The logarithmic function with base e is called the natural logarithmik function and it is denoted by ln: Thus With this new notation we have The logarithmic function with base 10 also is used frequently and there is a different notation for it: Instead of log 10 x we write log x. Thus With this notation

Change of Base.  For each a, b, x  ℝ with a, b > 0 and b ≠1, we have b x = ex ex ln b and These can be proved as follows bx bx = y  x ln b = y  y = e x b  b x = e x ln b. log b a = y  a = by by = ey ey ln b  y b = a Examples. Change base formulas can be given more generally as  For a, b, c, x  ℝ with a, b, c > 0, b ≠1, c we have

Logarithmic Equations. Example. undefined

Example. or We may assume x ≠ 1.1. Example.

Exponential Equations. Example.

Present Value. The present amount to be deposited as an investment to have a certain amount in the future. Simple Interest: A = P(1+rt)  P = A(1 + rt) -1 Compound Interest: Continuous Compound Interest : Example. If money earns 6 %, how much money should you deposit into an account so that it will accumulate to 1000 TL at the end of 10 years? Solution. We are given A = 1000, t = 10, r =0.06. We use simple interest formula Example. If money earns 6 % compounded annually, how much money should you deposit into an account so that it will accumulate to 1000 TL at the end of 10 years? Solution. A = 1000, t =10, m = 1, r = 0.06.

Example. If money earns 6 % compounded continuously, how much money should you deposit into an account so that it will accumulate to 1000 TL at the end of 10 years.? Solution. Substitute A = 1000, t = 1, r =0.06 in the formula P = A(1 + rt) -1 : Example (Doubling Time). How long will it take money to double if it is invested at a rate 7 % compounded continuously? Solution. In continuous compound interest formula, take A = 2P 2P, r =0.07  9.9 years

Example. How long will it take money to double if it is invested at a rate 7 % compounded annually? Solution. Take A = 2P 2P, m = 1, r =0.07 in the compound interest formula Example. A man has TL. He wants to deposit it into an account and use it in the future to buy a house. If he expects to have TL. in his account after 8 years, what must be the interest rate compounded continuously? Solution. Using the continuous compound interest formula  years.

Example. A man has TL. He wants to deposit it into an account at the rate of 8 % compounded continuously and use it in the future to buy a house. How long should he wait so that he will have TL. in his account? Solution. Using continuously compound interest formula years. Anuity. Any sequence of equal periodic payments. Computations inwolwing annuity are closely related to sums of finite geometric sequences: a, c ϵ ℝ ; n ϵ ℕ.ℕ. ac, ac 2,..., ac n-1 geometric sequence T=a+ac+ac ac n-1  cT=ac+ac ac n-1 +ac n  cT-T=ac n - a  (c - 1)T= a(c n -1)

Future Value of an Annuity. If the rate per compounding period is i and P TL is paid at the end of each time interval, then the value of this annuity at the end of n periods is a+ac+ac ac n-1 Example. What is the value of an annuity at the end of 3 years if 1000 TL is deposited at the end of every 6 months into an account earning 6 % compounded semiannually? TL If the payments are done in the beginning of each time interval, then we denote the annuity by A b. In that case

Example. What is the value of an annuity at the end of 3 years if 1000 TL is deposited in the beginning of every 6 months into an account earning 6 % compounded semiannually? TL Present Value of an Annuity. Let us denote by P the amount that must be deposited into an account at rate i per compounding period in order to be able to withdraw A TL at the end of each period so that after the n-th payment is made, no money will be left in the account. Then a+ac+ac ac n-1

Example. How much should you deposit into an account paying 6 % annually in order to be able to withdraw 200 TL at the end of every month for the next 5 years? TL Amortisation. z The process of paying a debt together with its interest in periodic payments is called amortzation. If the debt is P TL and it will be amortized in n equal payments at the rate i per compounding period, then the amount A that will be paid at the end of each period is obtained by considering the present value of the annuity:

Example. Assume that you buy a television set for 4200 TL and agree to pay for it 18 equal monthly payments at 1 % interest per month on the unpaid balance. How much are your payments? What is the total interest you will pay? TL Total interest is approximately 18(256.12) = – 4200 = TL.