Lesson 15 1 Defining and Evaluating a Logarithmic

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Lesson 15. 1: Defining and Evaluating a Logarithmic Function Review: An exponential function such

Lesson 15. 1: Defining and Evaluating a Logarithmic Function Review: An exponential function such as accepts values of the exponent as inputs, and delivers corresponding powers of 2 as outputs. What if we wanted to work backwards? If we want a function that takes in powers of 2 and delivers its corresponding exponent, we will need to explore the function’s reflection over the line y = x.

Graph and the line y = x Now lets reflect to graph the inverse

Graph and the line y = x Now lets reflect to graph the inverse of f(x) = 2 x

Graph and the line y = x But what should we call this inverse

Graph and the line y = x But what should we call this inverse function?

Logarithmic Functions The inverse of an exponential function is called a logarithmic function. A

Logarithmic Functions The inverse of an exponential function is called a logarithmic function. A logarithmic function accepts powers of a given base as inputs and delivers each input’s corresponding exponent as output. The function is written as where b is the base for which the inputs are powers.

Examples of Logarithmic Functions If f(x) = 2 x , then f-1(x) = log

Examples of Logarithmic Functions If f(x) = 2 x , then f-1(x) = log 2 x. If f(x) = 5 x , then f-1(x) = log 5 x. If f(x) = (¼)x , then f-1(x) = log¼ x. If f(x) = 10 x , then f-1(x) = log x (common logarithm) If f(x) = ex , then f-1(x) = loge x = ln x (natural logarithm)

Evaluating Logarithms Given , what is f(9), f(27), f(1/3)? f(9) = log 39 =

Evaluating Logarithms Given , what is f(9), f(27), f(1/3)? f(9) = log 39 = 2, since 32 = 9. f(27) = log 327 = 3, since 33 = 9. f(1) = log 31 = 0, since 30 = 1. f(1/3) = log 3(1/3) = -1, since 3 -1 = 1/3.

Converting Between Exponential and Logarithmic Equations

Converting Between Exponential and Logarithmic Equations

Example Fill in the following table:

Example Fill in the following table:

Example

Example

Evaluating Logarithms on a Calculator Your calculator can only evaluate common logarithms (base 10)

Evaluating Logarithms on a Calculator Your calculator can only evaluate common logarithms (base 10) and natural logarithms (base e). Use your calculator to evaluate the following. Round to the nearest hundredth: Log(25) = ? Ln(35) = ?

Make a Table and Graph f(x) = log 3 x x f(x) 1/9 1/3

Make a Table and Graph f(x) = log 3 x x f(x) 1/9 1/3 1 3 9

Make a Table and Graph f(x) = log 3 x x 1/9 1/3 1

Make a Table and Graph f(x) = log 3 x x 1/9 1/3 1 3 9 f(x) -2 -1 0 1 2