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Rewriting equations so all powers have the same base

Sometimes the common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property.

For example, consider the equation 256 = 4 x 5 . We can rewrite both sides of this equation as a power of 2. Then we apply the rules of exponents, along with the one-to-one property, to solve for x :

256 = 4 x 5 2 8 = ( 2 2 ) x 5 Rewrite each side as a power with base 2 . 2 8 = 2 2 x 10 Use the one-to-one property of exponents . 8 = 2 x 10 Apply the one-to-one property of exponents . 18 = 2 x Add 10 to both sides . x = 9 Divide by 2 .

Given an exponential equation with unlike bases, use the one-to-one property to solve it.

  1. Rewrite each side in the equation as a power with a common base.
  2. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form b S = b T .
  3. Use the one-to-one property to set the exponents equal.
  4. Solve the resulting equation, S = T , for the unknown.

Solving equations by rewriting them to have a common base

Solve 8 x + 2 = 16 x + 1 .

     8 x + 2 = 16 x + 1 ( 2 3 ) x + 2 = ( 2 4 ) x + 1 Write   8  and  16  as powers of   2.     2 3 x + 6 = 2 4 x + 4 To take a power of a power, multiply exponents .     3 x + 6 = 4 x + 4 Use the one-to-one property to set the exponents equal .             x = 2 Solve for  x .
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Solve 5 2 x = 25 3 x + 2 .

x = 1

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Solving equations by rewriting roots with fractional exponents to have a common base

Solve 2 5 x = 2 .

2 5 x = 2 1 2 Write the square root of  2 as a power of   2. 5 x = 1 2 Use the one-to-one property . x = 1 10 Solve for  x .
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Solve 5 x = 5 .

x = 1 2

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Do all exponential equations have a solution? If not, how can we tell if there is a solution during the problem-solving process?

No. Recall that the range of an exponential function is always positive. While solving the equation, we may obtain an expression that is undefined.

Solving an equation with positive and negative powers

Solve 3 x + 1 = −2.

This equation has no solution. There is no real value of x that will make the equation a true statement because any power of a positive number is positive.

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Solve 2 x = −100.

The equation has no solution.

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Solving exponential equations using logarithms

Sometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by taking the logarithm of each side. Recall, since log ( a ) = log ( b ) is equivalent to a = b , we may apply logarithms with the same base on both sides of an exponential equation.

Given an exponential equation in which a common base cannot be found, solve for the unknown.

  1. Apply the logarithm of both sides of the equation.
    • If one of the terms in the equation has base 10, use the common logarithm.
    • If none of the terms in the equation has base 10, use the natural logarithm.
  2. Use the rules of logarithms to solve for the unknown.

Solving an equation containing powers of different bases

Solve 5 x + 2 = 4 x .

           5 x + 2 = 4 x There is no easy way to get the powers to have the same base .          ln 5 x + 2 = ln 4 x Take ln of both sides .      ( x + 2 ) ln 5 = x ln 4 Use laws of logs .    x ln 5 + 2 ln 5 = x ln 4 Use the distributive law .    x ln 5 x ln 4 = 2 ln 5 Get terms containing  x  on one side, terms without  x  on the other . x ( ln 5 ln 4 ) = 2 ln 5 On the left hand side, factor out an  x .            x ln ( 5 4 ) = ln ( 1 25 ) Use the laws of logs .                    x = ln ( 1 25 ) ln ( 5 4 ) Divide by the coefficient of  x .
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Questions & Answers

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
ok, one moment
how do I post your graph for you?
it won't let me send an image?
also for the first one... y=mx+b so.... y=3x-2
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
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
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
strategies to form the general term
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
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
y=x will obviously be a straight line with a zero slope
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
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.
yes, correction on my end, I meant slope of 1 instead of slope of 0
what is f(x)=
Karim Reply
I don't understand
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."
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Now it shows, go figure?
what is this?
unknown Reply
i do not understand anything
lol...it gets better
I've been struggling so much through all of this. my final is in four weeks 😭
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
thank you I have heard of him. I should check him out.
is there any question in particular?
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Sure, are you in high school or college?
Hi, apologies for the delayed response. I'm in college.
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
I done know
What kind of answer is that😑?
I had just woken up when i got this message
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
i have a question.
how do you find the real and complex roots of a polynomial?
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
@Nare please let me know if you can solve it.
I have a question
hello guys I'm new here? will you happy with me
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
if not then how would I find it from a graph
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
you could also do it with two consecutive minimum points or x-intercepts
I will try that thank u
Case of Equilateral Hyperbola
Jhon Reply
f(x)=4x+2, find f(3)
f(3)=4(3)+2 f(3)=14
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
Practice Key Terms 1

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