4.6 Solve Exponential and Logarithmic Equations

1
4.6 Solve Exponential and Logarithmic Equations

• p. 267
• How do you use logs to solve an exponential
  equation?
• When is it easiest to use the definition of logs?
• Do you ever get a negative answer for logs?

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Exponential Equations

• One way to solve exponential equations is to use
  the property that if 2 powers w/ the same base
  are equal, then their exponents are equal.
• For bgt0 b?1 if bx by, then xy

3
Solve by equating exponents

• 43x 8x1
• (22)3x (23)x1 rewrite w/ same base
• 26x 23x3
• 6x 3x3
• x 1

Check ? 431 811 64 64
4
Your turn!

• 24x 32x-1
• 24x (25)x-1
• 4x 5x-5
• 5 x

Be sure to check your answer!!!
5
Solve the Equation
SOLUTION
Write original equation.
 
 
 
Rewrite 9 and 27 as powers with base 3.
 
Power of a power property
 
Property of equality for exponential equations
4x 3x 3
Property of equality for exponential equations
3
Solve for x.
The solution is 3.
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When you cant rewrite using the same base, you
can solve by taking a log of both sides

• 2x 7
• log22x log27
• x log27
• x 2.807

Use log2 because the x is on the 2 and log221
7
4x 15

• log44x log415
• x log415 log15/log4
• 1.95

Use change of base to solve
8
102x-34 21

• -4 -4
• 102x-3 17
• log10102x-3 log1017
• 2x-3 log 17
• 2x 3 log17
• x ½(3 log17)
• 2.115

9
5x2 3 25

• 5x2 22
• log55x2 log522
• x2 log522
• x (log522) 2
• (log22/log5) 2
• -.079

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Newtons Law of Cooling

• The temperature T of a cooling substance _at_ time t
  (in minutes) is
• T (T0 TR) e-rt TR
• T0 initial temperature
• TR room temperature
• r constant cooling rate of the substance

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• Youre cooking stew. When you take it off the
  stove the temp. is 212F. The room temp. is 70F
  and the cooling rate of the stew is r .046. How
  long will it take to cool the stew to a serving
  temp. of 100?

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• T0 212, TR 70, T 100 r .046
• So solve
• 100 (212 70)e-.046t 70
• 30 142e-.046t (subtract 70)
• .221 e-.046t (divide by 142)
• How do you get the variable out of the exponent?

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Cooling cont.

• ln .221 ln e-.046t (take the ln of both sides)
• ln .221 -.046t
• -1.556 -.046t
• 33.8 t
• about 34 minutes to cool!

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• How do you use logs to solve an exponential
  equation?
• Expand the logs to bring the exponent x down and
  solve for x.
• When is it easiest to use the definition of logs?
• When you have log information on the left equal
  to a number on the right.
• Do you ever get a negative answer for logs?
• Never! Logs are always positive.

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4.6 Assignment

• Page 271, 5-10, 14-21, 54-58

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Solve Exponential and Logarithmic Equations
4.6Day 2
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Solving Log Equations

• To solve use the property for logs w/ the same
  base
• Positive numbers b,x,y b?1
• If logbx logby, then x y

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log3(5x-1) log3(x7)

• 5x 1 x 7
• 5x x 8
• 4x 8
• x 2 and check
• log3(52-1) log3(27)
• log39 log39

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When you cant rewrite both sides as logs w/ the
same base exponentiate each side

• bgt0 b?1
• if x y, then bx by

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SOLUTION
Write original equation.
Exponentiate each side using base 4.
5x 1 64
5x 65
Add 1 to each side.
x 13
Divide each side by 5.
This is the way the book suggests you do the
problem.
21
Solve using the definition
 
Use the definition
 
 
 
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log5(3x 1) 2

• 52 (3x1) (use definition)
• 3x1 25
• x 8 and check
• Because the domain of log functions doesnt
  include all reals, you should check for
  extraneous solutions

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log5x log(x-1)2

• log (5x)(x-1) 2 (product property)
• log (5x2 5x) 2 (use definition)
• 5x2-5x 102
• 5x2 - 5x 100
• x2 x - 20 0 (subtract 100 and
  divide by 5)
• (x-5)(x4) 0 x5, x-4
• graph and youll see 5x is the only solution
  

2
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Solve the equation. Check for extraneous
solutions.
ln (7x 4) ln (2x 11)
SOLUTION
ln (7x 4) ln (2x 11)
Write original equation.
7x 4 2x 11
Property of equality for logarithmic equations
7x 2x 11 4
5x 15
Divide each side by 5.
x 3
25
Solve the equation. Check for extraneous
solutions.
log 5x log (x 1) 2
SOLUTION
log 5x log (x 5) 2
Write original equation.
log 5x(x 1) 2
Product property of logarithms
Use the definition
 
Distributive property
5x(x 1) 100
Subtract 100
 
Divide out a 5
 
 
Factor
 
Zero product property
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One More!log2x log2(x-7) 3

• log2x(x-7) 3
• log2 (x2- 7x) 3
• x2-7x 23
• x2 7x 8
• x2 7x 8 0
• (x-8)(x1)0
• x8 x -1

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Assignment 4.6 day 2

• p. 271, 26-42 all