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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

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
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.
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
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.
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
All real x except 5 and - 3
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
By using some imaginary no.
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
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
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
Please were did you get y=mx+b from
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.
thanks Tommy
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
"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?
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
consider r(a+b) = ra + rb. The a and b are the trig identity.
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?
Practice Key Terms 1

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