MATH 1314 – College Algebra – Logarithms
Definition: Let b be any positive number other than 1 and let x be any positive number. Then by
definition, the logarithm to the base b of x, denoted log b x , is the number that can be used as an
exponent on the base b in order to result in x.
Examples: According to the definition, log 2 8 is the exponent that can be used on the base 2 in
! 2 is 8, 3 is the exponent that can be used on the base 2
order to result in 8. Because the 3rd power of
to produce 8. So it follows that log 2 8 = 3 . Similarly,
!
log 3 9 = 2 because 32 = 9 ,
!
"2
1
1 ,
log10 100 = "2 because 10 = 12 = 100
10
!
2
2
log8 4 = ! because 8 2 / 3 = 3 8 = 2 2 = 4 , and
3
0
log 5 1 = 0 because
! 5 = 1.
!
!
Definitions:
! log b x is the base b logarithm function. Its domain is (0,") and it range is ("#,#) .
The domain is especially important. Only positive quantities have logarithms.
!
( )
( )
!
Example: Give the domain of f (x) = log 9 (2x " 3) .
The!quantity whose logarithm is indicated must be positive. !
So
2x " 3 > 0
2x > 3
!
x > 23
!
( )
!
!
The domain is the interval 23 ," .
!
Exercises: Determine the domain of each of the following.
1. f (x) = log 7 (5x + 10)
2. f (x) = log 9 (x 2 " 4)
3. f (x) = log 3 (5x 2 + 1)
!
Definitions: A base 10 logarithm, log10 x , is usually abbreviated log x and is called a common
logarithm of x. A base e logarithm, loge x , is usually abbreviate ln x and is called a natural
!
!
logarithm of x.
!
! of the following.
Exercises: Determine each
!
4. log 2 16
5. log 5 125
6. log 3 19
()
!
!
!
11. lne 3
( )
1
10. log 10
9. log1000
!
!
7. log 7 1
8. log 9 27
12. lne"2
13. log 7 7 3
y
! The statements log
Definitions:
! b x = y and b =!x are equivalent;!that means if one of the
statements is true so is the other.
log 3 t = 4 is true, then
! For example, if !
! it must also be true that
!
4
3 = t . Likewise, if p 5 = w , then log p w = 5 . A statement in the form logb x = y is said to be in
! a statement !
logarithmic form and
in the form b y = x is said to be in exponential form.
!
Exercises: Write the equivalent exponential statement for each of the following.
!
!
!
14. log 2 32 = 5
15. log 2 18 = "3
16. log p n = k
17. log x = 2
18. ln5 = w
!
()
(Thomason – Fall 2008)
!
!
!
!
!
Exercises: Write the equivalent logarithmic statement for each of the following.
1
19. 16 3/ 2 = 64
20. 9"2 = 81
21. d 0 = 1
22. 10 u = 37
23. e 5 = r
!
Definition: An equation that has the variable in an exponent is called an exponential equation.
Exponential equations can often be solved
by isolating!the exponential term
!
! on one side of the
!
equation and then writing the equivalent logarithmic statement.
( )
Example: Solve 5 2 4 x + 1 = 41 exactly for x.
( ) = 40 Subtract 1 from both sides of the given equation.
52
2 4 x = 8 Divide both sides by 5.
log 2 8 =!4 x Write the equivalent logarithmic statement.
3 = 4 x Evaluate log 2 8 , which is 3.
3 = x Divide both sides by 4.
4
!
!
!
!
Exercises:
! Solve each of the following exactly for x.
2x
24. 4 3 "108 = 0
25. 10 5 x+2 + 5 = 7
26. 4e 2x " 5 = 7
( )
!
!
!
)
Definition: An equation that has the variable in a logarithm is called a logarithmic equation.
Logarithmic equations can often be solved by isolating
the logarithmic term on one side of the
!
!
equation and then writing the equivalent exponential statement.
Exercises: Solve each of the following exactly for x.
! " 2) " 8 = 16
27. 8log 4 (5x
28. 3log(2x " 4) "1 = 5
!
!
(
Example: Solve 5log 2 (x " 3) " 7 = 13 exactly for x.
5log 2 (x " 3) = 20 Add 7 to both sides of the given equation.
log 2 (x " 3) = 4 Divide both sides by 5.
2 4 = x "!3 Write the equivalent exponential statement.
16 = x " 3 Evaluate 2 4 , which is 16.
19 = x Add 3 to both sides.
!
!
!
!
!
!
4x
29. 7ln(3x + 4) + 5 = 26
Property: The base b exponential function and the base b logarithmic function are inverse
functions. For example if we let f (x) = b x and g(x) = log
! b x , we see that
!
log b x
=x
( f o g)(x) = f (g(x)) = f (logb x) = b
and
x
x
(g o f )(x) = g( f (x)) = g(b
! ) = logb (b )!= x .
Examples: 3log 3 5 = 5
( )
log 5 5 23 = 23
!
!
7 log 7 p = p
(
)
log 4 4 x"3 = x " 3
!
!
(Thomason – Fall 2008)
!
10 log 2 = 2
2%
"
log 7 $ 7 5x ' = 5x 2
#
!&
!
e ln v = v
(
8 log 8 (2x+4) = 2x + 4
)
log 10 3x"2 = 3x " 2
!
!
" 2%
ln$e 3p ' = 3p 2
#
&