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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

a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
Practice Key Terms 1

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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