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We can express the relationship between logarithmic form and its corresponding exponential form as follows:

log b ( x ) = y b y = x , b > 0 , b 1

Note that the base b is always positive.

Because logarithm is a function, it is most correctly written as log b ( x ) , using parentheses to denote function evaluation, just as we would with f ( x ) . However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as log b x . Note that many calculators require parentheses around the x .

We can illustrate the notation of logarithms as follows:

Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means y = log b ( x ) and y = b x are inverse functions.

Definition of the logarithmic function

A logarithm    base b of a positive number x satisfies the following definition.

For x > 0 , b > 0 , b 1 ,

y = log b ( x )  is equivalent to  b y = x

where,

  • we read log b ( x ) as, “the logarithm with base b of x ” or the “log base b of x . "
  • the logarithm y is the exponent to which b must be raised to get x .

Also, since the logarithmic and exponential functions switch the x and y values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,

  • the domain of the logarithm function with base b   is   ( 0 , ) .
  • the range of the logarithm function with base b   is   ( , ) .

Can we take the logarithm of a negative number?

No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.

Given an equation in logarithmic form log b ( x ) = y , convert it to exponential form.

  1. Examine the equation y = log b x and identify b , y , and x .
  2. Rewrite log b x = y as b y = x .

Converting from logarithmic form to exponential form

Write the following logarithmic equations in exponential form.

  1. log 6 ( 6 ) = 1 2
  2. log 3 ( 9 ) = 2

First, identify the values of b , y , and x . Then, write the equation in the form b y = x .

  1. log 6 ( 6 ) = 1 2

    Here, b = 6 , y = 1 2 , and   x = 6. Therefore, the equation log 6 ( 6 ) = 1 2 is equivalent to 6 1 2 = 6 .

  2. log 3 ( 9 ) = 2

    Here, b = 3 , y = 2 , and   x = 9. Therefore, the equation log 3 ( 9 ) = 2 is equivalent to 3 2 = 9.

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Write the following logarithmic equations in exponential form.

  1. log 10 ( 1, 000, 000 ) = 6
  2. log 5 ( 25 ) = 2
  1. log 10 ( 1 , 000 , 000 ) = 6 is equivalent to 10 6 = 1 , 000 , 000
  2. log 5 ( 25 ) = 2 is equivalent to 5 2 = 25
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Converting from exponential to logarithmic form

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base b , exponent x , and output y . Then we write x = log b ( y ) .

Converting from exponential form to logarithmic form

Write the following exponential equations in logarithmic form.

  1. 2 3 = 8
  2. 5 2 = 25
  3. 10 4 = 1 10,000

First, identify the values of b , y , and x . Then, write the equation in the form x = log b ( y ) .

  1. 2 3 = 8

    Here, b = 2 , x = 3 , and y = 8. Therefore, the equation 2 3 = 8 is equivalent to log 2 ( 8 ) = 3.

  2. 5 2 = 25

    Here, b = 5 , x = 2 , and y = 25. Therefore, the equation 5 2 = 25 is equivalent to log 5 ( 25 ) = 2.

  3. 10 4 = 1 10,000

    Here, b = 10 , x = 4 , and y = 1 10,000 . Therefore, the equation 10 4 = 1 10,000 is equivalent to log 10 ( 1 10,000 ) = 4.

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Questions & Answers

how can are find the domain and range of a relations
austin Reply
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
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Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
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meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Practice Key Terms 3

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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