<< Chapter < Page Chapter >> Page >

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

Algebraic

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

4 3 v 2 = 4 v

Got questions? Get instant answers now!

64 4 3 x = 16

x = 1 3

Got questions? Get instant answers now!

2 3 n 1 4 = 2 n + 2

n = 1

Got questions? Get instant answers now!

36 3 b 36 2 b = 216 2 b

b = 6 5

Got questions? Get instant answers now!

( 1 64 ) 3 n 8 = 2 6

Got questions? Get instant answers now!

For the following exercises, use logarithms to solve.

e r + 10 10 = −42

No solution

Got questions? Get instant answers now!

8 10 p + 7 7 = −24

p = log ( 17 8 ) 7

Got questions? Get instant answers now!

7 e 3 n 5 + 5 = −89

Got questions? Get instant answers now!

e 3 k + 6 = 44

k = ln ( 38 ) 3

Got questions? Get instant answers now!

5 e 9 x 8 8 = −62

Got questions? Get instant answers now!

6 e 9 x + 8 + 2 = −74

x = ln ( 38 3 ) 8 9

Got questions? Get instant answers now!

e 2 x e x 132 = 0

x = ln 12  

Got questions? Get instant answers now!

7 e 8 x + 8 5 = −95

Got questions? Get instant answers now!

10 e 8 x + 3 + 2 = 8

x = ln ( 3 5 ) 3 8

Got questions? Get instant answers now!

8 e 5 x 2 4 = −90

no solution

Got questions? Get instant answers now!

e 2 x e x 6 = 0

x = ln ( 3 )

Got questions? Get instant answers now!

3 e 3 3 x + 6 = −31

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

4 + log 2 ( 9 k ) = 2

k = 1 36

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

10 4 ln ( 9 8 x ) = 6

x = 9 e 8

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

n = 1

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

No solution

Got questions? Get instant answers now!

log 4 ( 6 m ) = log 4 3 m

Got questions? Get instant answers now!

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

No solution

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

x = ± 10 3

Got questions? Get instant answers now!

For the following exercises, solve each equation for x .

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

Got questions? Get instant answers now!

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

x = 10

Got questions? Get instant answers now!

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

x = 0

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

x = 3 4

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

ln ( 3 x ) = 2

x = e 2 3 2.5

Graph of ln(3x)=y and y=2.
Got questions? Get instant answers now!

log ( 4 ) + log ( 5 x ) = 2

x = 5

Graph of log(4)+log(-5x)=y and y=2.
Got questions? Get instant answers now!

7 + log 3 ( 4 x ) = −6

Got questions? Get instant answers now!

ln ( 4 x 10 ) 6 = 5

x = e + 10 4 3.2

Graph of ln(4x-10)-6=y and y=-5.
Got questions? Get instant answers now!

log ( 4 2 x ) = log ( 4 x )

Got questions? Get instant answers now!

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).
Got questions? Get instant answers now!

ln ( 2 x + 9 ) = ln ( 5 x )

Got questions? Get instant answers now!

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).
Got questions? Get instant answers now!

log ( x 2 + 13 ) = log ( 7 x + 3 )

Got questions? Get instant answers now!

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).
Got questions? Get instant answers now!

ln ( x ) ln ( x + 3 ) = ln ( 6 )

Got questions? Get instant answers now!

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).
Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

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).
Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

e 5 x = 17 using the natural log

ln ( 17 ) 5 0.567

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

3 4 x 5 = 38 using the common log

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

Got questions? Get instant answers now!

50 e 0.12 t = 10 using the natural log

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

x 44655 . 7143

Got questions? Get instant answers now!

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)

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Extensions

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

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

Questions & Answers

can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
Abena Reply
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
Karim Reply
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
Practice Key Terms 1

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask