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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

how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
a²=4
Roy Reply
hil
Roy Reply
A bridge is to be built in the shape of a semi-elliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be 8 feet. Find the height of the arch at its center
Abdulfatah Reply
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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