5.3 Logarithmic Functions & Graphs Find common logarithms and natural logarithms with and without a calculator. Convert between exponential equations and logarithmic equations. Change logarithmic bases. Graph logarithmic functions. Solve applied problems involving logarithmic functions. Simplify expressions of the type logaax and

A logarithm is just an exponent. Our first question What is a logarithm ? A logarithm is just an exponent. Evaluate: log39

Logarithmic Function, Base b We define y = logb x as that number y such that x = by, where x > 0 and b is a positive constant other than 1. We read logb x as “the logarithm, base b, of x.” A logarithm is an exponent!   3

Finding Certain Logarithms - Example Find each of the following logarithms. log10 10,000 log10 0.01 log8 8 log216 log6 1 log2(1/4) 4

Basic Properties of Logarithms For b > 0, and b ≠ 1 Logb1 = 0 Logbb = 1 Logbbx = x Simplify

Basic Properties of Natural Logarithms Evaluate the following. ln (1) ln (e) ln (ex) = 0 = 1 = x = x

Logarithms Convert each of the following to a logarithmic equation. A logarithm is an exponent!   Convert each of the following to a logarithmic equation. a) 16 = 2x b) 10–3 = 0.001 c) et = 70 7

Example Rewrite in logarithmic form. 64 = 43 73 = x

Logarithms A logarithm is an exponent!   Convert each of the following to an exponential equation. a) log 2 32= 5 b) log b Q= 8 c) x = log t M 9

Example Rewrite in exponential form log416 = 2 n=logmx

Logarithms A base 10 logarithm is known as the common logarithm. We denote log10x = log x A base e logarithm is known as the Natural logarithm. We denote logex = ln x

Example Evaluate log 10000 log(1/1000) log 512 log(-10)

Example Find each of the following natural logarithms on a calculator. Round to four decimal places. a) log 645,778 b) ln 0.0000239 c) log (5) d) ln e e) ln 1 13

The Change-of-Base Formula For any logarithmic bases a and b, and any positive number M, Find log5 8 using common logarithms. Find log5 8 using natural logarithms. 14

Logarithmic Functions Log functions are inverses of exponential functions (y = 2x ). We can draw the graph of the inverse of an exponential function by interchanging x and y. (x = 2y ). 15

Graphs of Logarithmic Functions   , where b > 0 and b ≠ 1. Domain: (0, ∞) Range: (-∞, ∞) x-intercept is (1,0) No y-intercept The y-axis is a vertical asymptote (x = 0). f is 1-1 (one-to-one) Graph passes through the point (b, 1) If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.  

Graphs of Logarithmic Functions - Example Graph: y = f (x) = log5 x. Solution: y = log5 x is equivalent to x = 5y. Select y and compute x. 17

Example Describe how each graph can be obtained from the graph of y = ln x. Give the domain and the vertical asymptote of each function. a) f (x) = ln (x + 3) 18

Example f(x) = ln(x-4) + 2 y = -lnx - 2 For each function, state the transformations applied to y = lnx. Determine the vertical asymptote, and the domain and range for each function. f(x) = ln(x-4) + 2 y = -lnx - 2

Example Find the domain of each function algebraically.