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Solve 2 x = 3 x + 1 .

x = ln 3 ln ( 2 3 )

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Is there any way to solve 2 x = 3 x ?

Yes. The solution is 0.

Equations containing e

One common type of exponential equations are those with base e . This constant occurs again and again in nature, in mathematics, in science, in engineering, and in finance. When we have an equation with a base e on either side, we can use the natural logarithm    to solve it.

Given an equation of the form y = A e k t , solve for t .

  1. Divide both sides of the equation by A .
  2. Apply the natural logarithm of both sides of the equation.
  3. Divide both sides of the equation by k .

Solve an equation of the form y = Ae kt

Solve 100 = 20 e 2 t .

100 = 20 e 2 t 5 = e 2 t Divide by the coefficient of the power . ln 5 = 2 t Take ln of both sides . Use the fact that  ln ( x )  and  e x  are inverse functions . t = ln 5 2 Divide by the coefficient of  t .
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Solve 3 e 0.5 t = 11.

t = 2 ln ( 11 3 ) or ln ( 11 3 ) 2

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Does every equation of the form y = A e k t have a solution?

No. There is a solution when k 0 , and when y and A are either both 0 or neither 0, and they have the same sign. An example of an equation with this form that has no solution is 2 = −3 e t .

Solving an equation that can be simplified to the form y = Ae kt

Solve 4 e 2 x + 5 = 12.

4 e 2 x + 5 = 12 4 e 2 x = 7 Combine like terms . e 2 x = 7 4 Divide by the coefficient of the power . 2 x = ln ( 7 4 ) Take ln of both sides . x = 1 2 ln ( 7 4 ) Solve for  x .
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Solve 3 + e 2 t = 7 e 2 t .

t = ln ( 1 2 ) = 1 2 ln ( 2 )

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Extraneous solutions

Sometimes the methods used to solve an equation introduce an extraneous solution    , which is a solution that is correct algebraically but does not satisfy the conditions of the original equation. One such situation arises in solving when the logarithm is taken on both sides of the equation. In such cases, remember that the argument of the logarithm must be positive. If the number we are evaluating in a logarithm function is negative, there is no output.

Solving exponential functions in quadratic form

Solve e 2 x e x = 56.

e 2 x e x = 56 e 2 x e x 56 = 0 Get one side of the equation equal to zero . ( e x + 7 ) ( e x 8 ) = 0 Factor by the FOIL method . e x + 7 = 0  or  e x 8 = 0 If a product is zero, then one factor must be zero . e x = 7  or e x = 8 Isolate the exponentials . e x = 8 Reject the equation in which the power equals a negative number . x = ln 8 Solve the equation in which the power equals a positive number .
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Solve e 2 x = e x + 2.

x = ln 2

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Does every logarithmic equation have a solution?

No. Keep in mind that we can only apply the logarithm to a positive number. Always check for extraneous solutions.

Using the definition of a logarithm to solve logarithmic equations

We have already seen that every logarithmic equation log b ( x ) = y is equivalent to the exponential equation b y = x . We can use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic expression.

For example, consider the equation log 2 ( 2 ) + log 2 ( 3 x 5 ) = 3. To solve this equation, we can use rules of logarithms to rewrite the left side in compact form and then apply the definition of logs to solve for x :

Questions & Answers

root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
how do I attempted a trig number as a starter
Tumwe Reply
cos 18 ____ sin 72 evaluate
Het Reply
Practice Key Terms 1

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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