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
$\text{\hspace{0.17em}}t.$
Divide both sides of the equation by
$\text{\hspace{0.17em}}A.$
Apply the natural logarithm of both sides of the equation.
Divide both sides of the equation by
$\text{\hspace{0.17em}}k.$
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
$\text{\hspace{0.17em}}2=\mathrm{-3}{e}^{t}.$
Solving an equation that can be simplified to the form
y =
Ae^{
kt }
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.
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
$\text{\hspace{0.17em}}x:$
Questions & Answers
how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic.
Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation
of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15)
it's standard equation is x^2 + y^2/16 =1
tell my why is it only x^2? why is there no a^2?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations