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
$\text{\hspace{0.17em}}x:$
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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
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.
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