Logarithmic and Exponential Function

Good Essays
LOGARITHMIC AND EXPONENTIAL FUNCTIONS
Inverse relations
Exponential functions
Exponential and logarithmic equations
One logarithm
THE LOGARITHMIC FUNCTION WITH BASE b is the function y = logb x. b is normally a number greater than 1 (although it need only be greater than 0 and not equal to 1). The function is defined for all x > 0. Here is its graph for any base b.

Note the following:
• For any base, the x-intercept is 1. Why?
To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
The logarithm of 1 is 0. y = logb1 = 0.
• The graph passes through the point (b, 1). Why?
The logarithm of the base is 1. logbb = 1. • | The graph is below the x-axis -- the logarithm is negative -- for | | | 0 < x < 1. | | | Which numbers are those that have negative logarithms? |
Proper fractions. • | The function is defined only for positive values of x. | | | logb(−4), for example, makes no sense. Since b is always positive, no power of b can produce a negative number. |
• The range of the function is all real numbers.
• The negative y-axis is a vertical asymptote (Topic 18).
Example 1. Translation of axes. Here is the graph of the natural logarithm, y = ln x (Topic 20).

And here is the graph of y = ln (x − 2) -- which is its translation 2 units to the right.

The x-intercept has moved from 1 to 3. And the vertical asymptote has moved from 0 to 2.
Problem 1. Sketch the graph of y = ln (x + 3).

This is a translation 3 units to the left. The x-intercept has moved from 1 to −2. And the vertical asymptote has moved from 0 to −3.
Exponential functions
The exponential function with positive base b > 1 is the function y = bx.
It is defined for every real number x. Here is its graph:

There are two important things to note:
• The y-intercept is at (0, 1). For, b0 = 1.
• The negative x-axis is a horizontal asymptote.

You May Also Find These Documents Helpful

  • Good Essays

    EXPONENTIAL AND LOGARITHMIC FUNCTIONS I.EXPONENTIAL FUNCTION A. Definition An exponential function is a function defined by f(x) = ax , where a > 0 and a ≠ 1. The domain of the function is the set of real numbers and the range is the set of positive numbers. B. Evaluating Exponential Functions 1. Given: f(x) = 2x, find a. f(3) = ____ b. f(5) = _____ c. f(-2) = ______ d. f(-4) = ______ 2. Evaluate f(x) = ( 1)x if 2 a. x = 2 ____ b. x = 4 _____ c. x = -3 ______ d…

    • 904 Words
    • 4 Pages
    Good Essays
  • Satisfactory Essays

    Exponential and Logarithmic Functions * Verify that the natural logarithm function defined as an integral has the same properties as the natural logarithm function earlier defined as the inverse of the natural exponential function. Integrals of Exponential and Logarithmic Functions Function | Integral | lnx | x ∙ lnx - x + c | logx | (x ∙ lnx - x) / ln(10) + c | logax | x(logax - logae) + c | ex | ex+c | ek∙x | 1 / k ∙ ek∙x + c | ax | ax / lna + c | xn | 1 / (n+1) ∙ xn+1 +…

    • 361 Words
    • 2 Pages
    Satisfactory Essays
  • Satisfactory Essays

    MATH133 Unit 5: Exponential and Logarithmic Functions Individual Project Assignment: Version 2A Show all of your work details for these calculations. Please review this Web site to see how to type mathematics using the keyboard symbols. IMPORTANT: See Question 1 in Problem 2 below for special IP instructions. This is mandatory. Problem 1: Photic Zone Light entering water in a pond, lake, sea, or ocean will be absorbed or scattered by the particles in the water and its intensity, I, will be attenuated…

    • 1387 Words
    • 9 Pages
    Satisfactory Essays
  • Good Essays

    A function is a relation in which each element of the domain is paired with exactly one element in the range. Two types of functions are the exponential functions and the logarithmic functions. Exponential functions are the functions in the form of y = ax, where ''a'' is a positive real number, greater than zero and not equal to one. Logarithmic functions are the inverse of exponential functions, y = loga x, where ''a'' is greater to zero and not equal to one. These functions have certain differences…

    • 853 Words
    • 3 Pages
    Good Essays
  • Good Essays

    Solving Exponential and Logarithmic Equations Exponential Equations (variable in exponent position) 1. Isolate the exponential portion ( base exp onent ): Move all non-exponential factors or terms to the other side of the equation. 2. Take ln or log of each side of the equation. • Make sure to use ln if the base is “e”. Then remember that ln e = 1 . • Make sure to use log if the base is 10. • If the base is neither “e” nor “10”, use either ln or log, your choice.. 3. Bring the power (exponent)…

    • 494 Words
    • 2 Pages
    Good Essays
  • Satisfactory Essays

    Exponential Function

    • 2732 Words
    • 11 Pages

    Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. Tell whether the function y = 2( 5 ) shows growth or decay. Then graph the function. a. This is an exponential growth function. c. This is an exponential decay function. x b. This is an exponential growth function. d. This is an exponential growth function. ____ 2. Graph the inverse of the relation. Identify the domain and range of the inverse. x y −1 4 1 2 3 1 5 0 7 1 a. c…

    • 2732 Words
    • 11 Pages
    Satisfactory Essays
  • Good Essays

    Exponential function

    • 2230 Words
    • 9 Pages

    In mathematics, the exponential function is the function ex, where e is the number (approximately 2.718281828) such that the function ex is its own derivative.[1][2] The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease) in the dependent variable. The function is often written as exp(x), especially when it is impractical to write the independent variable as a superscript…

    • 2230 Words
    • 9 Pages
    Good Essays
  • Good Essays

    Lesson 08.11 Exponential and Logarithmic Functions Activity Materials: Bowl 100 dimes Laptop Geogebra Microsoft word Pencil Paper Procedure: Count the total number of pieces of candy, coins, or whatever object you have chosen and record this number in the chart shown below. Total Number of Objects Spill this object on the flat surface and count the number of objects which land face up and the number of objects which land face down. Record each number in the chart above in the row…

    • 561 Words
    • 3 Pages
    Good Essays
  • Satisfactory Essays

    Cartesian Coordinate System 4. Straight Lines 5. Functions and their Graphs 6. The Algebra of Functions 7. Functions and Math Models 8. Limits Test 1 9. The Derivative 10. Basic Rules of Differentiation 11. The Product and Quotient Rules 12. The Chain Rule 13. Marginal Functions in Economics 14. Higher-Order Derivatives 15. Implicit…

    • 338 Words
    • 3 Pages
    Satisfactory Essays
  • Good Essays

    APPLICATIONS OF EXPONENTIAL| AND| LOGARITHMIC FUNCTIONS| EARTHQUAKE WORD PROBLEMS: As with any word problem, the trick is convert a narrative statement or question to a mathematical statement. Before we start, let's talk about earthquakes and how we measure their intensity. In 1935 Charles Richter defined the magnitude of an earthquake to be where I is the intensity of the earthquake (measured by the amplitude of a seismograph reading taken 100 km from the epicenter of the…

    • 394 Words
    • 2 Pages
    Good Essays