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
$\text{\hspace{0.17em}}256={4}^{x-5}.\text{\hspace{0.17em}}$ We can rewrite both sides of this equation as a power of
$\text{\hspace{0.17em}}2.\text{\hspace{0.17em}}$ Then we apply the rules of exponents, along with the one-to-one property, to solve for
$\text{\hspace{0.17em}}x:$
This equation has no solution. There is no real value of
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ 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
$\text{\hspace{0.17em}}\mathrm{log}\left(a\right)=\mathrm{log}\left(b\right)\text{\hspace{0.17em}}$ is equivalent to
$\text{\hspace{0.17em}}a=b,$ 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
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
100•3=300
300=50•2^x
6=2^x
x=log_2(6)
=2.5849625
so, 300=50•2^2.5849625
and, so,
the # of bacteria will double every (100•2.5849625) =
258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.