# 6.6 Exponential and logarithmic equations  (Page 3/8)

 Page 3 / 8

Solve $\text{\hspace{0.17em}}{2}^{x}={3}^{x+1}.$

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

Is there any way to solve $\text{\hspace{0.17em}}{2}^{x}={3}^{x}?$

Yes. The solution is $0.$

## Equations containing e

One common type of exponential equations are those with base $\text{\hspace{0.17em}}e.\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ on either side, we can use the natural logarithm    to solve it.

Given an equation of the form $\text{\hspace{0.17em}}y=A{e}^{kt}\text{,}$ solve for $\text{\hspace{0.17em}}t.$

1. Divide both sides of the equation by $\text{\hspace{0.17em}}A.$
2. Apply the natural logarithm of both sides of the equation.
3. Divide both sides of the equation by $\text{\hspace{0.17em}}k.$

## Solve an equation of the form y = Ae kt

Solve $\text{\hspace{0.17em}}100=20{e}^{2t}.$

Solve $\text{\hspace{0.17em}}3{e}^{0.5t}=11.$

$t=2\mathrm{ln}\left(\frac{11}{3}\right)\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\mathrm{ln}{\left(\frac{11}{3}\right)}^{2}$

Does every equation of the form $\text{\hspace{0.17em}}y=A{e}^{kt}\text{\hspace{0.17em}}$ have a solution?

No. There is a solution when $\text{\hspace{0.17em}}k\ne 0,$ and when $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}2=-3{e}^{t}.$

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

Solve $\text{\hspace{0.17em}}4{e}^{2x}+5=12.$

Solve $\text{\hspace{0.17em}}3+{e}^{2t}=7{e}^{2t}.$

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

## 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 $\text{\hspace{0.17em}}{e}^{2x}-{e}^{x}=56.$

Solve $\text{\hspace{0.17em}}{e}^{2x}={e}^{x}+2.$

$x=\mathrm{ln}2$

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 $\text{\hspace{0.17em}}{\mathrm{log}}_{b}\left(x\right)=y\text{\hspace{0.17em}}$ is equivalent to the exponential equation $\text{\hspace{0.17em}}{b}^{y}=x.\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}{\mathrm{log}}_{2}\left(2\right)+{\mathrm{log}}_{2}\left(3x-5\right)=3.\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}x:$

bsc F. y algebra and trigonometry pepper 2
given that x= 3/5 find sin 3x
4
DB
remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
Will
Joeval
(x2-2x+8)-4(x2-3x+5)
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
master
Soo sorry (5±Root11* i)/3
master
Mukhtar
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
Jeffrey
Jeffrey
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey