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When does an extraneous solution occur? How can an extraneous solution be recognized?

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When can the one-to-one property of logarithms be used to solve an equation? When can it not be used?

The one-to-one property can be used if both sides of the equation can be rewritten as a single logarithm with the same base. If so, the arguments can be set equal to each other, and the resulting equation can be solved algebraically. The one-to-one property cannot be used when each side of the equation cannot be rewritten as a single logarithm with the same base.

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Algebraic

For the following exercises, use like bases to solve the exponential equation.

4 3 v 2 = 4 v

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64 4 3 x = 16

x = 1 3

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2 3 n 1 4 = 2 n + 2

n = 1

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36 3 b 36 2 b = 216 2 b

b = 6 5

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( 1 64 ) 3 n 8 = 2 6

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For the following exercises, use logarithms to solve.

e r + 10 10 = −42

No solution

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8 10 p + 7 7 = −24

p = log ( 17 8 ) 7

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7 e 3 n 5 + 5 = −89

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e 3 k + 6 = 44

k = ln ( 38 ) 3

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5 e 9 x 8 8 = −62

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6 e 9 x + 8 + 2 = −74

x = ln ( 38 3 ) 8 9

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e 2 x e x 132 = 0

x = ln 12  

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7 e 8 x + 8 5 = −95

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10 e 8 x + 3 + 2 = 8

x = ln ( 3 5 ) 3 8

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8 e 5 x 2 4 = −90

no solution

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e 2 x e x 6 = 0

x = ln ( 3 )

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3 e 3 3 x + 6 = −31

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For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation.

log ( 1 100 ) = −2

10 2 = 1 100

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For the following exercises, use the definition of a logarithm to solve the equation.

4 + log 2 ( 9 k ) = 2

k = 1 36

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2 log ( 8 n + 4 ) + 6 = 10

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10 4 ln ( 9 8 x ) = 6

x = 9 e 8

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For the following exercises, use the one-to-one property of logarithms to solve.

ln ( 10 3 x ) = ln ( 4 x )

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log 13 ( 5 n 2 ) = log 13 ( 8 5 n )

n = 1

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log ( x + 3 ) log ( x ) = log ( 74 )

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ln ( 3 x ) = ln ( x 2 6 x )

No solution

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log 4 ( 6 m ) = log 4 3 m

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ln ( x 2 ) ln ( x ) = ln ( 54 )

No solution

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log 9 ( 2 n 2 14 n ) = log 9 ( 45 + n 2 )

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ln ( x 2 10 ) + ln ( 9 ) = ln ( 10 )

x = ± 10 3

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For the following exercises, solve each equation for x .

log ( x + 12 ) = log ( x ) + log ( 12 )

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ln ( x ) + ln ( x 3 ) = ln ( 7 x )

x = 10

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ln ( 7 ) + ln ( 2 4 x 2 ) = ln ( 14 )

x = 0

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log 8 ( x + 6 ) log 8 ( x ) = log 8 ( 58 )

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ln ( 3 ) ln ( 3 3 x ) = ln ( 4 )

x = 3 4

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log 3 ( 3 x ) log 3 ( 6 ) = log 3 ( 77 )

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Graphical

For the following exercises, solve the equation for x , if there is a solution . Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.

log 9 ( x ) 5 = −4

x = 9

Graph of log_9(x)-5=y and y=-4.
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ln ( 3 x ) = 2

x = e 2 3 2.5

Graph of ln(3x)=y and y=2.
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log ( 4 ) + log ( 5 x ) = 2

x = 5

Graph of log(4)+log(-5x)=y and y=2.
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7 + log 3 ( 4 x ) = −6

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ln ( 4 x 10 ) 6 = 5

x = e + 10 4 3.2

Graph of ln(4x-10)-6=y and y=-5.
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log ( 4 2 x ) = log ( 4 x )

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log 11 ( 2 x 2 7 x ) = log 11 ( x 2 )

No solution

Graph of log_11(-2x^2-7x)=y and y=log_11(x-2).
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ln ( 2 x + 9 ) = ln ( 5 x )

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log 9 ( 3 x ) = log 9 ( 4 x 8 )

x = 11 5 2.2

Graph of log_9(3-x)=y and y=log_9(4x-8).
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log ( x 2 + 13 ) = log ( 7 x + 3 )

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3 log 2 ( 10 ) log ( x 9 ) = log ( 44 )

x = 101 11 9.2

Graph of 3/log_2(10)-log(x-9)=y and y=log(44).
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ln ( x ) ln ( x + 3 ) = ln ( 6 )

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For the following exercises, solve for the indicated value, and graph the situation showing the solution point.

An account with an initial deposit of $6,500 earns 7.25 % annual interest, compounded continuously. How much will the account be worth after 20 years?

about $ 27 , 710.24

Graph of f(x)=6500e^(0.0725x) with the labeled point at (20, 27710.24).
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The formula for measuring sound intensity in decibels D is defined by the equation D = 10 log ( I I 0 ) , where I is the intensity of the sound in watts per square meter and I 0 = 10 12 is the lowest level of sound that the average person can hear. How many decibels are emitted from a jet plane with a sound intensity of 8.3 10 2 watts per square meter?

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The population of a small town is modeled by the equation P = 1650 e 0.5 t where t is measured in years. In approximately how many years will the town’s population reach 20,000?

about 5 years

Graph of P(t)=1650e^(0.5x) with the labeled point at (5, 20000).
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Technology

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate x to 3 decimal places .

1000 ( 1.03 ) t = 5000 using the common log.

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e 5 x = 17 using the natural log

ln ( 17 ) 5 0.567

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3 ( 1.04 ) 3 t = 8 using the common log

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3 4 x 5 = 38 using the common log

x = log ( 38 ) + 5 log ( 3 )     4 log ( 3 ) 2.078

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50 e 0.12 t = 10 using the natural log

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For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

7 e 3 x 5 + 7.9 = 47

x 2.2401

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ln ( 3 ) + ln ( 4.4 x + 6.8 ) = 2

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log ( 0.7 x 9 ) = 1 + 5 log ( 5 )

x 44655 . 7143

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Atmospheric pressure P in pounds per square inch is represented by the formula P = 14.7 e 0.21 x , where x is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of 8.369 pounds per square inch? ( Hint : there are 5280 feet in a mile)

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The magnitude M of an earthquake is represented by the equation M = 2 3 log ( E E 0 ) where E is the amount of energy released by the earthquake in joules and E 0 = 10 4.4 is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing 1.4 10 13 joules of energy?

about 5.83

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Extensions

Use the definition of a logarithm along with the one-to-one property of logarithms to prove that b log b x = x .

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Recall the formula for continually compounding interest, y = A e k t . Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm.

t = ln ( ( y A ) 1 k )

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Recall the compound interest formula A = a ( 1 + r k ) k t . Use the definition of a logarithm along with properties of logarithms to solve the formula for time t .

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Newton’s Law of Cooling states that the temperature T of an object at any time t can be described by the equation T = T s + ( T 0 T s ) e k t , where T s is the temperature of the surrounding environment, T 0 is the initial temperature of the object, and k is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm.

t = ln ( ( T T s T 0 T s ) 1 k )

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Questions & Answers

root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
how do I attempted a trig number as a starter
Tumwe Reply
cos 18 ____ sin 72 evaluate
Het Reply
Practice Key Terms 1

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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