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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

how can are find the domain and range of a relations
austin Reply
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
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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