6.6 Exponential and logarithmic equations (Page 3/8)

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Solve $2^{x} = 3^{x + 1} .$

$x = \frac{\ln 3}{\ln (\frac{2}{3})}$

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Q&A

Is there any way to solve $2^{x} = 3^{x} ?$

Yes. The solution is $x = 0.$

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.

How To

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

Divide both sides of the equation by $A .$
Apply the natural logarithm of both sides of the equation.
Divide both sides of the equation by $k .$

Solve an equation of the form y = Ae ^kt

Solve $100 = 20 e^{2 t} .$

\begin{array}{l} 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 & = \frac{\ln 5}{2} & Divide by the coefficient of t . \end{array}

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Solve $3 e^{0.5 t} = 11.$

$t = 2 \ln (\frac{11}{3})$ or $\ln {(\frac{11}{3})}^{2}$

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Q&A

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

No. There is a solution when $k \neq 0,$ 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 $4 e^{2 x} + 5 = 12.$

\begin{array}{l} 4 e^{2 x} + 5 = 12 \\ 4 e^{2 x} = 7 & Combine like terms . \\ e^{2 x} = \frac{7}{4} & Divide by the coefficient of the power . \\ 2 x = \ln (\frac{7}{4}) & Take ln of both sides . \\ x = \frac{1}{2} \ln (\frac{7}{4}) & Solve for x . \end{array}

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Solve $3 + e^{2 t} = 7 e^{2 t} .$

$t = \ln (\frac{1}{\sqrt{2}}) = - \frac{1}{2} \ln (2)$

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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 $e^{2 x} - e^{x} = 56.$

\begin{array}{l} 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 \begin{matrix}  \end{matrix} & 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 . \end{array}

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Solve $e^{2 x} = e^{x} + 2.$

$x = \ln 2$

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Q&A

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 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

cool u

Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

hello

BenJay

Method

I am eliacin, I need your help in maths

Rood

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Sir

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Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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Source: OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3

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6.6 Exponential and logarithmic equations (Page 3/8)

Equations containing e

Solve an equation of the form y = Ae kt

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

Extraneous solutions

Solving exponential functions in quadratic form

Using the definition of a logarithm to solve logarithmic equations

Questions & Answers

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Solve an equation of the form y = Ae ^kt

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