Using the definition of a logarithm to solve logarithmic equations
For any algebraic expression
$\text{\hspace{0.17em}}S\text{\hspace{0.17em}}$ and real numbers
$\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}c,$ where
$\text{\hspace{0.17em}}b>0,\text{}b\ne 1,$
[link] represents the graph of the equation. On the graph, the
x -coordinate of the point at which the two graphs intersect is close to 20. In other words
$\text{\hspace{0.17em}}{e}^{3}\approx 20.\text{\hspace{0.17em}}$ A calculator gives a better approximation:
$\text{\hspace{0.17em}}{e}^{3}\approx \mathrm{20.0855.}$
Use a graphing calculator to estimate the approximate solution to the logarithmic equation
$\text{\hspace{0.17em}}{2}^{x}=1000\text{\hspace{0.17em}}$ to 2 decimal places.
Using the one-to-one property of logarithms to solve logarithmic equations
As with exponential equations, we can use the one-to-one property to solve logarithmic equations. The one-to-one property of logarithmic functions tells us that, for any real numbers
$\text{\hspace{0.17em}}x>0,$$S>0,$$T>0\text{\hspace{0.17em}}$ and any positive real number
$\text{\hspace{0.17em}}b,$ where
$\text{\hspace{0.17em}}b\ne 1,$
So, if
$\text{\hspace{0.17em}}x-1=8,$ then we can solve for
$\text{\hspace{0.17em}}x,$ and we get
$\text{\hspace{0.17em}}x=9.\text{\hspace{0.17em}}$ To check, we can substitute
$\text{\hspace{0.17em}}x=9\text{\hspace{0.17em}}$ into the original equation:
$\text{\hspace{0.17em}}{\mathrm{log}}_{2}\left(9-1\right)={\mathrm{log}}_{2}\left(8\right)=3.\text{\hspace{0.17em}}$ In other words, when a logarithmic equation has the same base on each side, the arguments must be equal. This also applies when the arguments are algebraic expressions. Therefore, when given an equation with logs of the same base on each side, we can use rules of logarithms to rewrite each side as a single logarithm. Then we use the fact that logarithmic functions are one-to-one to set the arguments equal to one another and solve for the unknown.
For example, consider the equation
$\text{\hspace{0.17em}}\mathrm{log}\left(3x-2\right)-\mathrm{log}\left(2\right)=\mathrm{log}\left(x+4\right).\text{\hspace{0.17em}}$ To solve this equation, we can use the rules of logarithms to rewrite the left side as a single logarithm, and then apply the one-to-one property to solve for
$\text{\hspace{0.17em}}x:$
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.