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Definition of Exponential Function

By Sue Perkins,2014-11-26 11:38
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Definition of Exponential Function

CRP 762 Handout Logarithm

    Source: http://www.sosmath.com/algebra/algebra.html (Section: Logarithms and Exponential

    Functions)

    Definition of Exponential Function

    xf(x)aa is denoted by, where, and x is any real The exponential function f with base a1

    number. The function value will be positive because a positive base raised to any power is

    xf(x)apositive. This means that the graph of the exponential function will be located in

    quadrants I and II.

    For example, if the base is 2 and x = 4, the function value f(4) will equal 16. A corresponding

    xf(x)2point on the graph of would be (4, 16).

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Definition of Logarithmic Function

    , we have >0 , and For x >0, aa1

    Since x > 0, the graph of the above function will be in quadrants I and IV.

Comments on Logarithmic Functions

    3 The exponential equation could be written in terms of a logarithmic equation as 464

    log(64)3. 4

    12 The exponential equation 5can be written as the logarithmic equation 25

    1log()2. 525

     Since logarithms are nothing more than exponents, you can use the rules of exponents

    with logarithms.

     Logarithmic functions are the inverse of exponential functions. For example if (4, 16) is a

    point on the graph of an exponential function, then (16, 4) would be the corresponding

    point on the graph of the inverse logarithmic function.

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     The two most common logarithms are called common logarithms and natural logarithms.

    Common logarithms have a base of 10, and natural logarithms have a base of e. On your

    calculator, the base 10 logarithm is noted by log, and the base e logarithm is noted by ln.

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    Properties of Logarithms

    log10Property 1: because . a

    0Example 1: In the equation, the base is 14 and the exponent is 0. Remember that 141

    a logarithm is an exponent, and the corresponding logarithmic equation is log10where the 0 is the exponent. 14

    110Example 2: In the equation, the base is and the exponent is 0. Remember that ()122

    a logarithm is an exponent, and the corresponding logarithmic equation is. log101

    2

    0Example 3: Use the exponential equation to write a logarithmic equation. The x1

    base x is greater than 0 and the exponent is 0. The corresponding logarithmic equation is

    log10. x

    loga1Property 2: because . a

    1Example 4: In the equation, the base is 3, the exponent is 1, and the answer is 3. 33

    Remember that a logarithm is an exponent, and the corresponding logarithmic equation is

    log31. 3

    1Example 5: In the equation , the base is 87, the exponent is 1, and the answer is 8787

    87. Remember that a logarithm is an exponent, and the corresponding logarithmic

    log871equation is . 87

    1ppExample 6: Use the exponential equation to write a logarithmic equation. If the

    logp1base p is greater than 0, then . p

    xxxlogaxProperty 3: because . aaa

    44Example 7: Since you know that 33, you can write the logarithmic equation with

    4log34base 3 as . 3

    44Example 8: Since you know that 1313, you can write the logarithmic equation with

    4log134base 13 as . 13

    2Example 9: Use the exponential equation 416to write a logarithmic equation with

    2base 4. You can convert the exponential equation 416to the logarithmic equation

    2log162log1624. Since the 16 can be written as , the equation can be written 442log42. 4

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Rules of Logarithms

    a be a positive number such that a does not equal 1, let n be a real number, and let u and v Let

    be positive real numbers.

    Log(uv)Log(u);Log(v)Logarithmic Rule 1: aaa

    uLogarithmic Rule 2: Log()Log(u)Log(v)aaav

    nLog(u)nLog(u)Logarithmic Rule 3: aa

Since logarithms are nothing more than exponents, these rules come from the rules of exponents.

    aLet be greater than 0 and not equal to 1, and let n and m be real numbers.

    nmn;mExponential Rule 1: aaa

    262;68Example: Let a = 5, n = 2, and m = 6. and 5*525*1562539062555390625

    nanmaExponential Rule 2: ma

    25252640.0016Example: Let a = 5, n = 2, and m = 6. and 550.00166156255

    nmnm(a)aExponential Rule 3:

    266(5)(25)244140625Example: Let a = 5, n = 2, and m = 6. and

    2*612 (5)(5)244140625

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Note: The above information is from http://www.sosmath.com/. This is good website for a quick

    catch of the Math you have forgotten. It has lots of examples. The only thing I don’t like about

    this website is that when you open it, it has some pop-ads.

     Here you can select the section you want. Algebra and Matrix

    Algebra may be the materials we need in our class.

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