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The exponential function f ( x ) = b x is one-to-one, with domain ( , ) and range ( 0 , ) . Therefore, it has an inverse function, called the logarithmic function with base b . For any b > 0 , b 1 , the logarithmic function with base b , denoted log b , has domain ( 0 , ) and range ( , ) , and satisfies

log b ( x ) = y if and only if b y = x .

For example,

log 2 ( 8 ) = 3 since 2 3 = 8 , log 10 ( 1 100 ) = −2 since 10 −2 = 1 10 2 = 1 100 , log b ( 1 ) = 0 since b 0 = 1 for any base b > 0 .

Furthermore, since y = log b ( x ) and y = b x are inverse functions,

log b ( b x ) = x and b log b ( x ) = x .

The most commonly used logarithmic function is the function log e . Since this function uses natural e as its base, it is called the natural logarithm    . Here we use the notation ln ( x ) or ln x to mean log e ( x ) . For example,

ln ( e ) = log e ( e ) = 1 , ln ( e 3 ) = log e ( e 3 ) = 3 , ln ( 1 ) = log e ( 1 ) = 0 .

Since the functions f ( x ) = e x and g ( x ) = ln ( x ) are inverses of each other,

ln ( e x ) = x and e ln x = x ,

and their graphs are symmetric about the line y = x ( [link] ).

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -3 to 4. The graph is of two functions. The first function is “f(x) = e to power of x”, an increasing curved function that starts slightly above the x axis. The y intercept is at the point (0, 1) and there is no x intercept. The second function is “f(x) = ln(x)”, an increasing curved function. The x intercept is at the point (1, 0) and there is no y intercept. A dotted line with label “y = x” is also plotted on the graph, to show that the functions are mirror images over this line.
The functions y = e x and y = ln ( x ) are inverses of each other, so their graphs are symmetric about the line y = x .

At this site you can see an example of a base-10 logarithmic scale.

In general, for any base b > 0 , b 1 , the function g ( x ) = log b ( x ) is symmetric about the line y = x with the function f ( x ) = b x . Using this fact and the graphs of the exponential functions, we graph functions log b for several values of b > 1 ( [link] ).

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from 0 to 4. The graph is of three functions. All three functions a log functions that are increasing curved functions that start slightly to the right of the y axis and have an x intercept at (1, 0). The first function is “y = log base 10 (x)”, the second function is “f(x) = ln(x)”, and the third function is “y = log base 2 (x)”. The third function increases the most rapidly, the second function increases next most rapidly, and the third function increases the slowest.
Graphs of y = log b ( x ) are depicted for b = 2 , e , 10 .

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 1 , and r is any real number, then

1. log b ( a c ) = log b ( a ) + log b ( c ) (Product property) 2. log b ( a c ) = log b ( a ) log b ( c ) (Quotient property) 3. log b ( a r ) = r log b ( a ) (Power property)

Solving equations involving exponential functions

Solve each of the following equations for x .

  1. 5 x = 2
  2. e x + 6 e x = 5
  1. Applying the natural logarithm function to both sides of the equation, we have
    ln 5 x = ln 2 .

    Using the power property of logarithms,
    x ln 5 = ln 2 .

    Therefore, x = ln 2 / ln 5 .
  2. Multiplying both sides of the equation by e x , we arrive at the equation
    e 2 x + 6 = 5 e x .

    Rewriting this equation as
    e 2 x 5 e x + 6 = 0 ,

    we can then rewrite it as a quadratic equation in e x :
    ( e x ) 2 5 ( e x ) + 6 = 0 .

    Now we can solve the quadratic equation. Factoring this equation, we obtain
    ( e x 3 ) ( e x 2 ) = 0 .

    Therefore, the solutions satisfy e x = 3 and e x = 2 . Taking the natural logarithm of both sides gives us the solutions x = ln 3 , ln 2 .
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Solve e 2 x / ( 3 + e 2 x ) = 1 / 2 .

x = ln 3 2

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Solving equations involving logarithmic functions

Solve each of the following equations for x .

  1. ln ( 1 x ) = 4
  2. log 10 x + log 10 x = 2
  3. ln ( 2 x ) 3 ln ( x 2 ) = 0
  1. By the definition of the natural logarithm function,
    ln ( 1 x ) = 4 if and only if e 4 = 1 x .

    Therefore, the solution is x = 1 / e 4 .
  2. Using the product and power properties of logarithmic functions, rewrite the left-hand side of the equation as
    log 10 x + log 10 x = log 10 x x = log 10 x 3 / 2 = 3 2 log 10 x .

    Therefore, the equation can be rewritten as
    3 2 log 10 x = 2 or log 10 x = 4 3 .

    The solution is x = 10 4 / 3 = 10 10 3 .
  3. Using the power property of logarithmic functions, we can rewrite the equation as ln ( 2 x ) ln ( x 6 ) = 0 .
    Using the quotient property, this becomes
    ln ( 2 x 5 ) = 0 .

    Therefore, 2 / x 5 = 1 , which implies x = 2 5 . We should then check for any extraneous solutions.
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Questions & Answers

why n does not equal -1
K.kupar Reply
ask a complete question if you want a complete answer.
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
Darnell Reply
proof the formula integration of udv=uv-integration of vdu.?
Bg Reply
Find derivative (2x^3+6xy-4y^2)^2
Rasheed Reply
no x=2 is not a function, as there is nothing that's changing.
Vivek Reply
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
Joys Reply
y=800
Gift
800
Bg
how do u factor the numerator?
Drew Reply
Nonsense, you factor numbers
Antonio
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
Antonio
well as long as you got there eventually
zach
volume between cone z=√(x^2+y^2) and plane z=2
Kranthi Reply
answer please?
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
How do we find the horizontal asymptote of a function using limits?
Lerato Reply
Easy lim f(x) x-->~ =c
Antonio
solutions for combining functions
Amna Reply
what is a function? f(x)
Jeremy Reply
one that is one to one, one that passes the vertical line test
Andrew
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
Antonio
is x=2 a function?
The
restate the problem. and I will look. ty
jon Reply
is x=2 a function?
The
What is limit
MaHeSh Reply
it's the value a function will take while approaching a particular value
Dan
don ger it
Jeremy
what is a limit?
Dlamini
it is the value the function approaches as the input approaches that value.
Andrew
Thanx
Dlamini
Its' complex a limit It's a metrical and topological natural question... approaching means nothing in math
Antonio
is x=2 a function?
The
3y^2*y' + 2xy^3 + 3y^2y'x^2 = 0 sub in x = 2, and y = 1, isolate y'
Andrew Reply
what is implicit of y³+x²y³=5 at (2,1)
Estelita Reply
tel mi about a function. what is it?
Jeremy
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
moe
x or y those not matter is just to represent.
Bg
Practice Key Terms 7

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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