The exponential function
$f\left(x\right)={b}^{x}$ is one-to-one, with domain
$(\text{\u2212}\infty ,\infty )$ and range
$\left(0,\infty \right).$ Therefore, it has an inverse function, called the
logarithmic function with base$b.$ For any
$b>0,b\ne 1,$ the logarithmic function with base
b , denoted
${\text{log}}_{b},$ has domain
$(0,\infty )$ and range
$\left(\text{\u2212}\infty ,\infty \right),$ and satisfies
${\text{log}}_{b}\left(x\right)=y\phantom{\rule{0.2em}{0ex}}\text{if and only if}\phantom{\rule{0.2em}{0ex}}{b}^{y}=x.$
The most commonly used logarithmic function is the function
${\text{log}}_{e}.$ Since this function uses natural
$e$ as its base, it is called the
natural logarithm . Here we use the notation
$\text{ln}(x)$ or
$\text{ln}\phantom{\rule{0.1em}{0ex}}x$ to mean
${\text{log}}_{e}\left(x\right).$ For example,
and their graphs are symmetric about the line
$y=x$ (
[link] ).
At this
site you can see an example of a base-10 logarithmic scale.
In general, for any base
$b>0,b\ne 1,$ the function
$g\left(x\right)={\text{log}}_{b}(x)$ is symmetric about the line
$y=x$ with the function
$f\left(x\right)={b}^{x}.$ Using this fact and the graphs of the exponential functions, we graph functions
${\text{log}}_{b}$ for several values of
$b>1$ (
[link] ).
Before solving some equations involving exponential and logarithmic functions, let’s review the basic properties of logarithms.
Rule: properties of logarithms
If
$a,b,c>0,b\ne 1,$ and
$r$ is any real number, then
Now we can solve the quadratic equation. Factoring this equation, we obtain
$\left({e}^{x}-3\right)\left({e}^{x}-2\right)=0.$
Therefore, the solutions satisfy
${e}^{x}=3$ and
${e}^{x}=2.$ Taking the natural logarithm of both sides gives us the solutions
$x=\text{ln}\phantom{\rule{0.1em}{0ex}}3,\text{ln}\phantom{\rule{0.1em}{0ex}}2.$
The solution is
$x={10}^{4\text{/}3}=10\sqrt[3]{10}.$
Using the power property of logarithmic functions, we can rewrite the equation as
$\text{ln}\left(2x\right)-\text{ln}\left({x}^{6}\right)=0.$ Using the quotient property, this becomes
$\text{ln}\left(\frac{2}{{x}^{5}}\right)=0.$
Therefore,
$2\text{/}{x}^{5}=1,$ which implies
$x=\sqrt[5]{2}.$ We should then check for any extraneous solutions.
You can factorize the numerator of an expression. What's the problem there? here's an example.
f(x)=((x^2)-(y^2))/2
Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y).
so the overall function becomes :
((x+y)(x-y))/2
The
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
Antonio
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
A function it's a law, that for each value in the domaon associate a single one in the codomain
Antonio
function is a something which another thing depends upon to take place.
Example
A son depends on his father.
meaning here is
the father is function of the son.
let the father be y and the son be x. the we say
F(X)=Y.
Bg
yes the son on his father
pascal
a function is equivalent to a machine. this machine makes x to create y. thus, y is dependent upon x to be produced. note x is an independent variable