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Log Exponential

Definition (Exponential Functions) If is a real number with and then the function is an exponential function with base

Example (Applications of the Growth and Decay Functions)

(a) If is invested for years at compounded quarterly, the interest earned is What is the interest earned after 15 years.

    Solution. The interest earned is

(b) The percent concentration of a certain drug in the bloodstream at any time is given by the equation Graph this equation for

    Solution. There is no intercept in this domain, the horizontal asymptote is the line the intercept is and the graph is

Definition (Growth Functions) A function of the form where and   is called a growth function. The domain is the set of all real numbers and the range is The asymptote is the -axis (negative half ) and the intercept is

Definition (Decay Functions) A function of the form where and   is called a decay function. The domain is the set of all real numbers and the range is The asymptote is the  is the -axis (positive half ) and the intercept is

Example (Supply) If the supply function for a product is given by where represents the number of hundreds of units, what will be the price when the producers are willing to supply 600 units?

    Solution. We have

Example (Total Cost) If the total cost function for a product is given by where is the number of items produced, what is the total cost of producing 30 units?

    Solution. We have,

Example (Total Revenue) If the demand function for a product is given by where is the price per unit when units are demanded, what is the total revenue when 40 units are demanded and supplied?

    Solution. The price per unit is so when 40 units are supplied the price is per unit. So the total revenue is

Example (Compound Interest) If is invested at compounded continuously, the future value at any time (in years) is given by (a) What is the amount after year? (b) How long before the investment doubles?

    Solution. After one year, we have The investment doubled when and so we solve for in We have, or 7 years.

Example (Consumer Price Index) By using data from the U.S. Bureau of Labor Statistics for the years 1968-2000, the purchasing power of a 1983 dollar can be modeled with the function where is the number of years past 1960. (a) Find ,      and for each, write a sentence that interprets its meaning. (b) How long before it will cost to purchase goods that cost in 1983?

    Solution. We have and As years go by the power of the $1 of 1983 loses its power. This can also be seen from the graph of :



We want to solve for in the equation to find out when the is to purchase goods that cost in 1983. We have, or 52.5 years. The year will be

Definition (Simple Interest and Future Value) If a sum of money (called the principal) is invested for a period of time at an interest rate per period, the simple interest is given by the formula: and the future value of the investment is

Example (Future Value for Simple Interest) If $21,200 is invested at an annual simple interest rate of 5%, what is the future value of the investment after 2 years?
    
    Solution. The future value is given by the formula and since and   we have
    

Example (Interest for Simple Interest) If $7,700 is invested for 5 years at an annual simple interest rate of 15%, how much interest is earned?
    
    Solution. The interest earned is where and so we have

Example (Principal for Simple Interest) A firm buys 15 file cabinets at $166.23 each, with the bill due in 90 days. How much must the firm deposit now to have enough to pay the bill if money is worth 6% per year? Use 360 days in a year.
    
    Solution. The future value is We are looking for the principal, and We use the formula and we have and solving for we get

Example (Doubling Time for Simple Interest) If $5000 is invested at 8% annual simple interest, how long does it take to double to $10,000?
    
    Solution. The future value is given by the formula and we are given a value of We are asked to find when and We have
    






years.

Definition (Periodic Compounding Interest) If dollars is invested for years at a nominal interest rate compounded times per year, then the total number of compounded periods is and the interest rate per period is and the future value is or

Example (Future Value for Compounding Periodically) Find the future value if $3500 is invested for 6 years at 8% compounded quarterly.

    Solution. The future value is given by the formula where and so we have
    

Example (Interest for Compounding Periodically) Find the interest that will be earned if $5000 is invested for 3 years at 10% compounded semiannually.

    Solution. The interest earned is the future value minus the principal. So we find the future value first. The future value is given by where and so we have
    


Therefore, the interest earned is

Example (Principal for Compounding Periodically) What present value amounts to $100,000 if it is invested for 10 years at 8% compounded quarterly?

    Solution. The present value can be found using the formula where the future value and so we have

Example (Graphing of an Exponential Function) Plot the functions by either using a graphing calculator or if you can use transformations for:

(a)

(b)  


(c)

Examples (Exponential Functions)

(1) Solve the following exponential equations:

    (a)
    
    (b)
    
    (c)
    
    (d)

(2) Sketch the graph of the following functions. Label any intercepts and asymptotes.

    (a)
    
    (b)
    
    (c)
    
    (d)
    

Solutions (Exponential Functions)

(1) Solve the following exponential equations:

(a)
    
    Solution. We find that,
    
        
        
Therefore,
    
(b)
    
    Solution.  We find that,
    
        
        
Therefore,
    
(c)
    
    Solution. We find that
    
    
    
Solving the quadrant equation we obtain:
    
    
(2) Sketch the graph of the following functions. Label any intercepts and asymptotes.

(a)
    
    Solution. The graph can be obtained by plotting or using and then applying the absolute value. We have,
    


The graph as no vertical asymptotes, the horizontal asymptote is there are no -intercepts, the -intercept is
    
(b)
    
    Solution. The graph can be obtained by plotting or using and then applying and the horizontal shift left by 1. We have,
    


The graph as no vertical asymptotes, the horizontal asymptote is there are no -intercepts, the -intercept is

(c)
    
    Solution. The graph can be obtained by using , , and then applying the function   at each point. We have,
    


The graph as no asymptotes the and intercept is

Definition (Logarithmic Functions) For and the logarithmic function has domain base and is defined by The expressions (logarithmic form) and (exponential form) are equivalent. The is called the base in both and and the in is the logarithm and the in is the exponent. Thus a logarithm is an exponent.

Example (Converting Between Exponential and Logarithmic Forms) Convert to exponential form.

    Solution.  We have   

Example (Converting Between Exponential and Logarithmic Forms) Convert to logarithmic form.

    Solution. We have

Definition (Common Logarithmic Function) The logarithmic function with base 10 (common logarithmic function) is usually denoted by instead of .

Definition (Natural Logarithmic Function)  The logarithmic function with base (natural logarithmic function) is usually denoted by instead of .

Example (Graphs of Logarithmic Functions) Using basic transformations of functions we use a basic graph and apply different transformations to obtain the graph of         
    Solution. The graph of can be obtained by graphing and applying the scaling factor of
    

Example (Graphs of Logarithmic Functions) Using basic transformations of functions we use a basic graph and apply different transformations to obtain the graph of     

    Solution. The graph of can be obtained by graphing and applying a horizontal shift left 4 units and then scaling with the factors of and

Example (Graphs of Logarithmic Functions) Using basic transformations of functions we use a basic graph and apply different transformations to obtain the graph of

    Solution. The graph of can be obtained by graphing and then reflecting through the -axis.  

Proposition (Properties of Logarithms) If then

    (i)     for all real numbers         
    
    (ii)
    
    (iii)         
    
    (iv) all real numbers such that

    (v) for all real numbers and with

    (vi)   for all real numbers and with

    (vii) all real numbers such that
    

Example (Properties of Logarithms) Write as the sum or difference of logarithms for

    Solution. We have

Example (Properties of Logarithms) Write as a sum or difference of logarithms for

    Solution. We have

Example (Properties of Logarithms) Write as one logarithm.

    Solution. We have

Example (Properties of Logarithms) Write the expression as the sum or difference of two logarithmic functions containing no exponents for

    Solution. We have

Proposition (Change of Base Formula) If then



and is called the change of base formula.

Example (Change of Base Formula) Use the change of base to evaluate

    Solution. We have,

Example (Change of Base Formula) Use the change of base formula to graph

    Solution. We use and we see that scales the graph and we have,

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