College Algebra: Lesson 3 Exponential and Logarithmic Functions

Topics Topics include exponential functions logarithmic functions properties of logarithms exponential and logarithmic equations modeling with exponential and logarithmic functions

Exponential Functions Defined by f(x) = bx where b > 0 and b ≠ 1 Domain consists of all real numbers. Range consists of all positive numbers. One-to-one function. Inverse is a function.

Exponential Functions Graphs pass through (0, 1). Y-intercept is 1. There is no x-intercept. Graph approaches, but does not touch, the x-axis. If b > 1, graph goes up to the right. The greater the value of b, the steeper the increase of the graph. If 0 < b < 1, graph goes down to the right. The smaller the value of b, the steeper the decrease.

The Natural Base e e is e is called the natural base. an irrational number defined as the value that (1 + 1/n)n approaches as n →∞ ≈ 2.718281827 e is called the natural base. The function f(x) = ex is called the natural exponential function.

Graphs of Exponential Functions b > 1 0 < b < 1 Graphs of Exponential Functions

Formulas for Compound Interest n compoundings per year Continuous compounding A = P(1 + r/n)nt A = Pert Formulas for Compound Interest

Logarithmic Functions For x > 0 and b > 0, b ≠ 1, y = logb x is equivalent to by = x. The domain of f(x) = logbx consists of all positive real numbers. The range of f(x) = logbx consists of all real numbers.

Logarithmic Functions The graphs of all logarithmic functions pass through (1, 0). The x-intercept is 1. There is no y-intercept. The graph of f(x) = logb x approaches, but does not touch, the y-axis. If b > 1, the graph of f(x) = logb x goes up to the right. If 0 < b < 1, the graph of f(x) = logb x goes down to the right.

Special Logarithms The logarithmic function with base 10. Common Logarithms Natural Logarithms The logarithmic function with base 10. f(x) = log10 x is usually written f(x) = log x The logarithmic function with base e is expressed as ln x. read “el en of x.” Special Logarithms

Graphs of Logarithmic Functions b > 1 0 < b < 1 Graphs of Logarithmic Functions

Graphs of Exponential and Logarithmic Functions The graphs of an exponential function (green) and its inverse, a logarithmic function (red), are reflections about y = x.

Examples Write log5125 = 3 in exponential form. 53=125 Log to exponential Exponential to log Write log5125 = 3 in exponential form. 53=125 Write √36 = 6 in logarithmic form. √36 = 6 can be rewritten as 361/2 = 6. In logarithmic form, this would be log366= 1/2 Examples

Basic Logarithmic Properties Involving 1 logb b = 1 since b1 = b logb 1 = 0 since b0 = 1

Inverse Properties of Logarithms Exponential The logarithm with base b of b raised to a power equals that power: logb bx = x b raised to the logarithm with base b of a number equals that number: blogbx = x Inverse Properties of Logarithms

Properties of Natural Logarithms General Logarithms Natural Logarithms logb 1 = 0 logb b = 1 logb bx = x blogbx = x ln 1 = 0 ln e = 1 ln ex = x eln x = x Properties of Natural Logarithms

More Properties of Logarithms Product Rule Quotient Rule The logarithm of a product is the sum of the logarithms. The logarithm of a quotient is the difference of the logarithms. More Properties of Logarithms

Still More Properties of Logarithms The logarithm of a number with an exponent is the product of the exponent and the logarithm of that number.

Examples log4(64x5) log464 + log4x5 log464 + 5log4x 3 + 5log4x Expand

Examples 6logx + 2logy ln7 – 3lnx logx6 + logy2 ln7 – lnx3 Condense

Change-of-Base Property The logarithm of M (any positive number) with base b is equal to the logarithm of M with any new base divided by the logarithm of b with that new base.

Example Use a calculator to evaluate log1571 to four decimal places. Problem Solution Use a calculator to evaluate log1571 to four decimal places. Apply change of base property. Let a = 10, b = 15, M = 71 Example

Exponential Equations An exponential equation is an equation containing a variable in an exponent. Some exponential equations can be solved by expressing each side of the equation as a power of the same base.

Example Solve 3x-2 = 9x+4 3x-2 = 32(x+4) = 32x+8 Problem Solution Solve 3x-2 = 9x+4 We can rewrite the equation in the form bM = bN and set M = N. 3x-2 = 32(x+4) = 32x+8 x – 2 = 2x + 8 -10 = x or x = -10 Example

Solving Exponential Equations Most exponential equations cannot be rewritten so that each side has the same base.

Solving Exponential Equations Isolate the exponential expression. Take the natural log on both sides of the equation for bases of other than 10. Use the common log for base 10. Simplify using one of these properties. ln bx = x ln b ln ex = x log 10x = x Solve for the variable.

Example Solve 5x = 1.4 ln5x = ln1.4 xln5 = ln1.4 x = (ln1.4)/(ln5) Problem Solution Solve 5x = 1.4 Isolate the exponential expression. Take the natural log on both sides. Simplify using ln bx=x ln b or ln ex = x ln5x = ln1.4 xln5 = ln1.4 x = (ln1.4)/(ln5) Example

Example Solve 400e0.005x = 1600 400e0.005x = 1600 e0.005x = 4 Problem Solution Solve 400e0.005x = 1600 Isolate the exponential expression. Take the natural log on both sides of the equation. 400e0.005x = 1600 e0.005x = 4 ln e0.005x = ln 4 0.005x = ln 4 x = ln 4/0.005 Example

Example Solve e2x – 6ex + 5 = 0. e2x – 6ex + 5 = 0 Problem Solution Solve e2x – 6ex + 5 = 0. This equation is quadratic in form. It can be factored. Set each factor equal to zero and solve. e2x – 6ex + 5 = 0 (ex - 5)(ex – 1) = 0 ex – 5 = 0 ex = 5 ln ex = ln 5 x = ln 5 ex – 1 = 0 ex = 1 x = 0 {0, ln 5} Example

Logarithmic Equations A logarithmic equation is an equation containing a variable in a logarithmic expression. Some logarithmic equations can be expressed in the form logbM = c. These can be solved by rewriting in exponential form.

Logarithmic Equations Remember— logarithmic expressions are defined only for logarithms of positive real numbers. Always check proposed solutions of a logarithmic equation in the original equation. Exclude any proposed solutions that produce the log of a negative number or the log of zero.

Example Solve log6(4x – 1) = 3 Rewrite in exponential form: Problem Solution Solve log6(4x – 1) = 3 Rewrite in exponential form: 63 = 4x – 1 216 = 4x – 1 217 = 4x 54.25 = x Example

Solving Logarithmic Equations Express the equation in the form logb M = logb N (single logarithm with coefficient of 1.) Use the one-to-one property to rewrite the equation without logarithms. If logb M = logb N, then M = N. Solve for the variable. Check the proposed solutions in the original equation.

Example Solve 2ln(3x) = 8 Remember 2ln (3x) = 8 ln (3x) = 4 e4 = 3x Problem Solution Solve 2ln(3x) = 8 Remember eln x = x So eln 3x = 3x 2ln (3x) = 8 ln (3x) = 4 e4 = 3x e4/3 = x Example

Example Solve logx + log(x + 15) = 2 log(x(x + 15)) = 2 Problem Solution Solve logx + log(x + 15) = 2 log(x(x + 15)) = 2 log(x2 + 15x) = 2 102 = x2 + 15x 100 = x2 + 15x x2 + 15x - 100 = 0 (x + 20)(x – 5) = 0 x = 5, -20 {5} -20 results in a negative log Example

Problem Solution Solve ln(x – 4) – ln(x + 1) = ln6 Example

Exponential Growth and Decay F(t) = A = A0ekt k > 0 Graph rises from left to right. F(t) = A = A0ekt k < 0 Graph falls from left to right. Exponential Growth and Decay

Problem Solution The function A = 82.3e-0.002t models the population of Germany, A, in millions t years after 2003. What was the population of Germany in 2003? A = 82.3e-0.002(0) A = 82.3e0 A = 82.3 million Application

Problem Solution The function A = 82.3e-0.002t models the population of Germany, A, in millions t years after 2003. Is the population of Germany increasing or decreasing? Decreasing, since the exponent representing the growth rate is negative. Application

Problem Solution The function A = 82.3e-0.002t models the population of Germany, A, in millions t years after 2003. In what year will the population of Germany be 81.5 million? 81.5=82.3e-0.002t 0.99 = e-0.002t ln0.99 = lne-0.002t -0.002t = ln0.99 2008 Application