Logarithms
Table of Contents
If
x> 0
and there is a constant
b≠1
, then
y=logbx
, then if and only if
by=x
In the equation
logbx
, y is referred to as the logarithm, b is the base, and x is the argument.
The notation is read “the logarithm (or log) base b of x .” The definition of a logarithm indicates that a logarithm is an exponent
y=logbx
is the logarithmic form of
by=x by=x by=x
is the exponential form of
y=logbx
Example
1 Write each equation in its exponential form.
3=log10〖(x+8)〗c.
log5125=x
2 Write the following in its logarithmic form:
x=〖25〗(1/2)
Solution:
Use the definition
y=logbx
if and only if
by=x
(a)
2=log7x
if and only if
72=x
(b).
3=log10〖(x+8)〗
if and only if
〖10〗3=x+8
(c)
log5125=x
if and only if
5x=125
y=logbx
x=〖25〗(1/2) if and only if 1/2=log25x
If b is a positive real number
(b≠1)
such that
bx=by, then x=y
Example:
evaluate
log232=x
Solution: Use the definition
y=logbx
if and only if
by=x x=log232
if and only if
2x=32
⇒
2x=25
⇒ Thus, by Equality of Exponents,
x=5
b≠1
If b, a, and c are positive real numbers,
, and n is a real number, then:
Examples
(a)
2logbx+1/2logb〖(x+4)〗
(b)
4logb〖(x+2)〗–3logb〖(x–5)〗
(a)
logb〖(xy2)〗
(b)
logb〖(x2√y)/Z5〗
Solutions: 1. Use the properties of logarithms to rewrite each expression as a single logarithm:
(a)
2logbx+1/2logb〖(x+4)〗=logb〖x2〗+logb〖〖(x+4)〗(1/2)〗
power property
=logb〖[x2〖(x+4)〗(1/2)〗]
product property
(b)
4logb〖(x+2)〗–logb(x–5)=logb〖(x+2)4〗–〖3log〗b〖〖(x+4)〗3〗
power property
=logb〖〖(x+2)〗4/〖(x–5)〗3〗
quotient property
(a)
logb〖xy2〗=logbx+logb〖y2〗
product property ⇒
logb〖xy2〗=logbx+logb〖y2〗
(b)
logb〖(x2√y)/Z5〗 =logb〖(x2√y)〗–logb〖Z5〗
quotient property
=logb〖x2+〗logb√y–logb〖Z5〗
product property
=〖2logb〗〖x+〗logb√y–5logbZ
power property
A common logarithm has a base of 10. If there is no base given explicitly, it is common. You can easily find common logs of powers often. You can use your calculator to evaluate common logs.
A natural logarithm has a base of e. We write natural logarithms as ln. In other words,
logex=lnx
.
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