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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

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.

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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