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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

the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
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
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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
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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