<< Chapter < Page Chapter >> Page >

Solve 2 x = 3 x + 1 .

x = ln 3 ln ( 2 3 )

Got questions? Get instant answers now!

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 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve 3 e 0.5 t = 11.

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

Got questions? Get instant answers now!

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 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve 3 + e 2 t = 7 e 2 t .

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

Got questions? Get instant answers now!

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 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve e 2 x = e x + 2.

x = ln 2

Got questions? Get instant answers now!

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

1KI POWER 1/3 PLEASE SOLUTIONS
Prashant Reply
hii
Amit
how are you
Dorbor
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin
the least possible degree ?
Dejen Reply
(1+cosA)(1-cosA)=sin^2A
BINCY Reply
good
Neha
why I'm sending you solved question
Mirza
Teach me abt the echelon method
Khamis
exact value of cos(π/3-π/4)
Ankit Reply
Practice Key Terms 1

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask