Base-e Exponential and Logarithmic Functions Section 11.5 Base-e Exponential and Logarithmic Functions

Objectives Define the natural exponential function Graph the natural exponential function Use base-e exponential formulas and functions in applications Define base-e logarithms Evaluate natural logarithmic expressions Graph the natural logarithmic function Use base-e logarithmic formulas and functions in applications

Objective 1: Define the Natural Exponential Function Of all possible bases for an exponential function, e is the most convenient for problems involving growth or decay. Since these situations occur often in natural settings, we call ƒ(x) = ex the natural exponential function. The function defined by ƒ(x) = ex is the natural exponential function (or the base-e exponential function) where e = 2.71828. . . . The domain of ƒ(x) = ex is the interval (–∞, ∞). The range is the interval (0, ∞).

Objective 2: Graph the Natural Exponential Function To graph ƒ(x) = ex, we construct a table of function values by choosing several values for x and finding the corresponding values of ƒ(x). For example, x = –2, we have Substitute -2 for each x. Use a calculator. On a scientific calculator, press: 2+– 2nd ex. Round to the nearest tenth. We enter (–2, 0.1) in the table. Similarly, we find ƒ(–1), ƒ(0), ƒ(1) and ƒ(2), enter each result in the table, and plot the ordered pairs.

Objective 2: Graph the Natural Exponential Function

Objective 2: Graph the Natural Exponential Function The graphs of many functions are translations of the natural exponential function.

Objective 3: Use Base-e Exponential Formulas and Functions in Applications If a quantity P increases or decreases at an annual rate r, compounded continuously, the amount A after t years is given by A = Pert Read as “A equals P times e to the rt power.”

EXAMPLE 2 City Planning. The population of a city is currently 15,000, but economic conditions are causing the population to decrease 3% each year. If this trend continues, find the population in 30 years. Strategy We will substitute 15,000 for P, –0.03 for r, and 30 for t in the formula A = Pert and calculate the value of A. Why Since the population is decreasing 3% each year, the annual growth rate is –3%, or –0.03.

EXAMPLE 2 Solution In 30 years, the expected population will be 6,099. City Planning. The population of a city is currently 15,000, but economic conditions are causing the population to decrease 3% each year. If this trend continues, find the population in 30 years. Solution In 30 years, the expected population will be 6,099.

Objective 4: Define Base-e Logarithms Of all possible bases for a logarithmic function, e is the most convenient for problems involving growth or decay. Since these situations occur often in natural settings, base-e logarithms are called natural logarithms or Napierian logarithms after John Napier (1550–1617). They are usually written as ln x rather than loge x: In general, the logarithm of a number is an exponent. For natural logarithms, ln x is the exponent to which e is raised to get x. Translating this statement into symbols, we have e ln x = x

Objective 5: Evaluate Natural Logarithmic Expressions Evaluate each natural logarithmic expression:

EXAMPLE 4 Evaluate each natural logarithmic expression: Strategy Since the base is e in each case, we will ask “To what power must e be raised to get the given number?” Why That power is the value of the logarithmic expression.

EXAMPLE 4 Evaluate each natural logarithmic expression: Solution

Objective 6: Graph the Natural Logarithmic Function The natural logarithmic function with base e is defined by the equations ƒ(x) = ln x or y = ln x where ln x = loge x The domain of ƒ(x) = ln x is the interval (0, ∞), and the range is the interval (–∞, ∞).

Objective 6: Graph the Natural Logarithmic Function To graph ƒ(x) = ln x, we can construct a table of function values, plot the resulting ordered pairs, and draw a smooth curve through the points to get the graph shown. The natural exponential function and the natural logarithm function are inverse functions. The figure shows that their graphs are symmetric to the line y = x.

Objective 7: Use Base-e Logarithmic Formulas and Functions in Applications If a population grows exponentially at a certain annual rate, the time required for the population to double is called the doubling time. Formula for Doubling Time If r is the annual rate, compounded continuously, and t is the time required for a population to double, then

EXAMPLE 6 Doubling Time. The population of the Earth is growing at the approximate rate of 1.133% per year. If this rate continues, how long will it take for the population to double? Strategy We will substitute 1.133% expressed as a decimal for r in the formula for doubling time and evaluate the right side using a calculator. Why We can use this formula because we are given the annual rate of continuous compounding.

EXAMPLE 6 Doubling Time. The population of the Earth is growing at the approximate rate of 1.133% per year. If this rate continues, how long will it take for the population to double? Solution Since the population is growing at the rate of 1.133% per year, we substitute 0.0133 for r in the formula for doubling time and simplify. Don’t forget to substitute the decimal form of 1.133%, which is 0.0133, for r. 0.0133 Use a calculator. On a scientific calculator, press 2 LN  .0033 = . Round to the nearest year. At the current growth rate, the population of the Earth will double in about 52 years.