Rewriting equations so all powers have the same base
Sometimes the
common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property.
For example, consider the equation
We can rewrite both sides of this equation as a power of
Then we apply the rules of exponents, along with the one-to-one property, to solve for
Given an exponential equation with unlike bases, use the one-to-one property to solve it.
Rewrite each side in the equation as a power with a common base.
Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form
Use the one-to-one property to set the exponents equal.
Solve the resulting equation,
for the unknown.
Solving equations by rewriting them to have a common base
Do all exponential equations have a solution? If not, how can we tell if there is a solution during the problem-solving process?
No. Recall that the range of an exponential function is always positive. While solving the equation, we may obtain an expression that is undefined.
Solving an equation with positive and negative powers
Solve
This equation has no solution. There is no real value of
that will make the equation a true statement because any power of a positive number is positive.
Sometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by taking the logarithm of each side. Recall, since
is equivalent to
we may apply logarithms with the same base on both sides of an exponential equation.
Given an exponential equation in which a common base cannot be found, solve for the unknown.
Apply the logarithm of both sides of the equation.
If one of the terms in the equation has base 10, use the common logarithm.
If none of the terms in the equation has base 10, use the natural logarithm.
Use the rules of logarithms to solve for the unknown.
Solving an equation containing powers of different bases
the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement
A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon
