Download 6.4 Logarithmic Functions

Sullivan , 8th ed.
MAC1140/MAC1147
6-4: Logarithmic Functions
The inverse function of y = f(x) = ax is x = ay, which is equivalent to the logarithmic
function y = log a x. This equivalence is used to convert between logarithmic and
exponential functions. If the base is the number e, then x = e y if and only if y = ln x.
Write the equation in logarithmic form:
Write in exponential form:
2
2.2
1. 16 = 4
2. e = M
3. logb 4 = 2
4. ln x = 4
Evaluate; if au = av, then u = v:
5. log5 3 25
6. ln e3
7. Find a so that the graph of f(x) = logax.
contains the point (32, 8); simplify.
y = loga x
or x = ay
a>1
Domain
Range
x-intercept
[0, )
(-, )
(1, 0)
0<a<1
[0, )
(-, )
(1, 0)
Vertical
asymptote
y-axis as y
-
y-axis as y

Characteristic
Increasing,
one-to-one
Decreasing
one-to-one
Passes through
(1, 0)
(a, 1)
(1, 0)
(a, 1)
Graph the given logarithmic function and its inverse; state the domain, range, and any vertical asymptote:
8. f(x) = 3 – ln (x + 2)
x
e3-y - 2
0
1
2
3
4
e(3
- x)
9. f(x) = 2 + log1/2 x
-2
=y
x=
(1/2)y-2
-1
0
1
2
3
(1/2)x-2
=y
Sullivan , 8th ed.
MAC1140/MAC1147
6.5: Properties of Logarithms
Properties: (1) loga 1  0 and loga a  1 ; (2) a loga M  M and log a a r  r ;
M
 log a M  log a N ; (5) log a M r  r log a M ;
(3) loga  MN   loga M  loga N ; (4) log a
N
(6) log a  1    log a N . If M, N, and a are positive real numbers and a 0 and b0, then (7) if M =
N 
N, loga M  loga N , and (8) if loga M  loga N , M = N.
Change of base formulas: (9) log a M  log b M and (10) log a M  ln M .
log b a
ln a
Use the properties of logarithms to find the exact value of each expression. Do not use a
calculator:
1. ln e
log7 15log7 3
2. log 6 9  log 6 4
2
3. 7
Use properties of logarithms to rewrite the logarithm in terms of p, q r, or s:
4. If ln 4 = p,
5. If ln 10 = q and ln 18 = r,
6. If ln5 = r and ln 225 = s,
ln 256 =
ln1.8 =
ln 3 45 =
Write as the sum or difference of logarithms:
2
5
7. logb 3 x y6
2
z
8.
  x  4 2  3
ln  2

 x  1 
Express as a single logarithm:
9. logb 2x  3 logb x  logb y 
10. 35 log 4 7 2x  2 log 4  5x   3 log 4 3
Sullivan , 8th ed.
MAC1140/MAC1147
Evaluate; round your answer to nearest hundredth.
11. log5 18
13. log5 343 . log725
12.
log 2
14. log24 . log4 6 . log6 8
Express y as a function of x. C is a positive number.
15. ln y = 15x + lnC
1
3
1
5
16. 5ln(y) = ln( x  5)  ln  x  7   lnC
Sullivan , 8th ed.
MAC1140/MAC1147
Try These (6.4-6.5)
Use the properties of logarithms to find the exact value of each expression. Do not use a
calculator:
1.
Graph the function: y = -log(x – 3) + 2
Domain: {xx > 3}
Range: (-, )
Asymptotes: x = 3
2. a. log816 – log82
log88 = 1
b. e
ln 29
x
x = 29
Write as the sum or difference of logarithms:
x3 x  1
1
3.
log
 x  2
2
 3log x 
2
log( x  1)  2log( x  2)
Express as a single logarithm:
4.
log
  x  3  x  1

x2  2 x  3
x2  7 x  6
x2

l
og
 log 


2
x 4
x2
  x  2  x  2   x  6  x  1 
  x  3  x  1


 x  3  x  1 
1
 log 

  log 

 x  2
 x  6  x  1 

  x  2  x  6  x  1 
Express y as a function of x. The constant C is a positive number.
5. log(y + 8) = 3x + log C
log (y + 8) – log C = 3x
y8
log
 3x
C
y8
 103 x
C
y + 8 = 103xC
y = 103xC - 8