CRP 762 Handout Logarithm
Source: http://www.sosmath.com/algebra/algebra.html (Section: Logarithms and Exponential
Definition of Exponential Function
xf(x)？aa is denoted by, where, and x is any real The exponential function f with base a，1
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).
Definition of Logarithmic Function
, we have >0 , and For x >0, aa，1
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 4？64
1；2？ The exponential equation 5can be written as the logarithmic equation ？25
？ Since logarithms are nothing more than exponents, you can use the rules of exponents
？ 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.
？ 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.
Properties of Logarithms
log1？0Property 1: because . a
0Example 1: In the equation, the base is 14 and the exponent is 0. Remember that 14？1
a logarithm is an exponent, and the corresponding logarithmic equation is log1？0where 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. log1？01
0Example 3: Use the exponential equation to write a logarithmic equation. The x？1
base x is greater than 0 and the exponent is 0. The corresponding logarithmic equation is
loga？1Property 2: because . a
1Example 4: In the equation, the base is 3, the exponent is 1, and the answer is 3. 3？3
Remember that a logarithm is an exponent, and the corresponding logarithmic equation is
1Example 5: In the equation , the base is 87, the exponent is 1, and the answer is 87？87
87. Remember that a logarithm is an exponent, and the corresponding logarithmic
log87？1equation is . 87
1p？pExample 6: Use the exponential equation to write a logarithmic equation. If the
logp？1base p is greater than 0, then . p
xxxloga？xProperty 3: because . a？aa
44Example 7: Since you know that 3？3, you can write the logarithmic equation with
4log3？4base 3 as . 3
44Example 8: Since you know that 13？13, you can write the logarithmic equation with
4log13？4base 13 as . 13
2Example 9: Use the exponential equation 4？16to write a logarithmic equation with
2base 4. You can convert the exponential equation 4？16to the logarithmic equation
2log16？2log16？24. Since the 16 can be written as , the equation can be written 442log4？2. 4
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: aa？a
262;68Example: Let a = 5, n = 2, and m = 6. and 5*5？25*15625？3906255？5？390625
nan；m？aExponential Rule 2: ma
25252；6；4？？0.0016Example: Let a = 5, n = 2, and m = 6. and 5？5？0.00166156255
nmnm(a)？aExponential Rule 3:
266(5)？(25)？244140625Example: Let a = 5, n = 2, and m = 6. and
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.