Download There a two types of logarithmic, one is Exponential Function

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There a two types of logarithmic, one is Exponential Function, another
one is Logarithmic Function
Definition of Exponential Function:
The exponential function f with base a is denoted by
, where
,
and x is any real number. The function value will be positive because a positive base
raised to any power is positive. 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 point on the graph of
Definition of Logarithmic Function:
For x >0, a>0 , and
, we have
would be (4, 16).
Since x > 0, the graph of the above function will be in quadrants I and IV.
Basic Properties
Take a > 0 and not equal to 1 . Since the exponential function
f : R -> R : x -> ax
are either increasing or decreasing, the inverse function is defined. This inverse
function is called the logarithmic function with base a. We write
loga (x)
So,
y
loga(x) = y <=> a = x
From this we see that the domain of the logarithmic function is the set of strictly positive
real numbers, and the range is R.
Example:
log2(8) = 3 ;
log3(sqrt(3)) = 0.5 ;
From the definition it follows immediately that
for x > 0
we have
aloga(x) = x
for all x
we have
loga(ax) = x
and
The graph below, it is helpful to "decode" a logarithm by saying "x is equal to b to the
power y." This is equivalent to saying "y is the base-b logarithm of x."
The expression y = logb x is a logarithmic function, with domain restrictions x > 0 and x
1. Any exponential function y = bx, b 1 and b > 0 has an inverse y = logb x. The graphs
of these functions are shown in Figures 1 and 2. Figure 1 shows a graph using a base b
greater than one, and Figure 2 displays a graph with a base b less than one but still
positive.
Figure 1, b > 1
Figure 2, 0 < b < 1
Using the inverse function property of logarithms, exponential expressions can be written
as logarithmic expressions and logarithmic expressions can be written as exponential
expressions. With a little practice, it's possible to convert between the two quite easily.
Consider the examples in this table:
Exponential form
Logarithmic Form
32 = 9
log3 9 = 2
2
log10 100 = 2
2
5 = 25
log 5 25 = 2
e = 20.0855
loge 20.0855 = 3
Note that logex can be
written as ln x
ln 20.0855 = 3
10 = 100
3
Properties of Logarithms
If R > 0, S > 0, b 1 and a is any real
number, then
logb RS = logb R + logb
log of a product
S
logbR/S = logb R - logb S log of a quotient
logb (Ra) = a logb R
log of a power
logb b = 1
If logbR = logb S then R = S
Examples
Simplify:
log 2x + log (x - 3)
= log (2x(x - 3))
= log (2x2 - 6x)
Simplify:
2 log x - log ( x + 1)
2
= log x - log (x + 1)
2
= log (x /(x+1) )
Solve for x:
3 log x = log 64
3
log x = log 64
x3 = 64
x=4
Solve for x:
2 log x = log 9 + log 8 - log 18
log x2 = log 9 + log 8 - log 18
2
log x = log (9)(8) - log 18
2
log x = log ((9*8)/18)
log x2 = log 4
x2 = 4
x=2
Use a calculator to solve for x.
x
10 = 2275
x
log 10 = log 2275
x log 10 = log 2275
x = log 2275
x = 3.36
Use a calculator to solve for x.
x
8 = 640
x
log 8 = log 640
x log 8 = log 640
x = (log 640)/(log 8)
x = 3.31