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LOGARITHMS

Shubham Jaiswal
22/04/2017 0 0
Definition and Basics of Logarithms--Part II

Saying   that    logaM =x  means exactly the same thing as saying ax = M .

In other words:

 

logais the number to which you raise a in order to get M.

 

Keep this in mind in thinking about logarithms.  It makes lots of things obvious.

For example:  What is log2 8?  Ask yourself "To what power should I raise 2 in order to get 8?"  Since 8 is 2the answer is "3."  So  log28=3. 

Here's another way that remembering the rule:

 

"logais the number to which you raise a in order to get M."

 

can make some things almost obvious.  For example, what is  2 log2 5?   Note that log2is the power to which  2 is being raised. 

 But log2 is the number to which you raise 2 in order to get 5!  So if you raise 2 to that number you get 5!!  In other words 

2 log2 5 = 5.

Let's use 

 

"logais the number to which you raise a in order to get M,"

 

to understand logarithms of product.  For example:  What is log2(8*32)?  

Notice that 8=23 and 32=2so 8*32=232 = 23+5 =2.  

But this means  that 

log2(8*32)=log2(28) = 8 = 3+5=

log2(23)+log2(25)=log2(8)+log2(32)

In other words, the log of the product 8*32 equals  the sum of the logs of 8 and 32.

Of course there is nothing special about the base 2.  The same idea holds for other logarithms.

Apply this idea to the following examples: 


 

If log264=7 and log2256=8  then log2(64*256) =

 

Rules

1. Inverse properties:   loga ax = x   and   a(loga x) = x

2. Product:  loga (xy) = loga x + loga y

3. Quotient:  

4. Power:   loga (xp) = p loga x

5. Change of base formula:

Careful!!

loga (x + y) ≠ loga x + loga y

loga (x – y) ≠ loga x loga y

Note: ln x is sometimes written Ln x or LN x

 

Logarithm Rules

The base b logarithm of a number is the exponent that we need to raise the base in order to get the number.

Logarithm definition

When b is raised to the power of y is equal x:

b y = x

Then the base b logarithm of x is equal to y:

logb(x) = y

For example when:

24 = 16

Then

log2(16) = 4

Logarithm as inverse function of exponential function

The logarithmic function,

y = logb(x)

is the inverse function of the exponential function,

x = by

So if we calculate the exponential function of the logarithm of x (x>0),

f (f -1(x)) = blogb(x) = x

Or if we calculate the logarithm of the exponential function of x,

f -1(f (x)) = logb(bx) = x

Natural logarithm (ln)

Natural logarithm is a logarithm to the base e:

ln(x) = loge(x)

When e constant is the number:

e=\lim_{x\rightarrow \infty }\left ( 1+\frac{1}{x} \right )^x = 2.718281828459...

or

e=\lim_{x\rightarrow 0 }\left ( 1+ \right x)^\frac{1}{x}

 

See: Natural logarithm

Inverse logarithm calculation

The inverse logarithm (or anti logarithm) is calculated by raising the base b to the logarithm y:

x = log-1(y) = b y

Logarithmic function

The logarithmic function has the basic form of:

f (x) = logb(x)

Logarithm rules

Rule name Rule
Logarithm product rule
logb(x ? y) = logb(x) + logb(y)
Logarithm quotient rule
logb(x / y) = logb(x) - logb(y)
Logarithm power rule
logb(x y) = y ? logb(x)
Logarithm base switch rule
logb(c) = 1 / logc(b)
Logarithm base change rule
logb(x) = logc(x) / logc(b)
Derivative of logarithm
f (x) = logb(x) ⇒ f ' (x) = 1 / ( x ln(b) )
Integral of logarithm
∫ logb(x) dx = x ? ( logb(x) - 1 / ln(b) ) + C
Logarithm of negative number
logb(x) is undefined when x≤ 0
Logarithm of 0
logb(0) is undefined
\lim_{x\to 0^+}\textup{log}_b(x)=-\infty
Logarithm of 1
logb(1) = 0
Logarithm of the base
logb(b) = 1
Logarithm of infinity
lim logb(x) = ∞,when x→∞

See: Logarithm rules

 

Logarithm product rule

The logarithm of the multiplication of x and y is the sum of logarithm of x and logarithm of y.

logb(x ? y) = logb(x) + logb(y)

For example:

log10(3 ? 7) = log10(3) + log10(7)

Logarithm quotient rule

The logarithm of the division of x and y is the difference of logarithm of x and logarithm of y.

logb(x / y) = logb(x) - logb(y)

For example:

log10(3 / 7) = log10(3) - log10(7)

Logarithm power rule

The logarithm of x raised to the power of y is y times the logarithm of x.

logb(x y) = y ? logb(x)

For example:

log10(28) = 8? log10(2)

Logarithm base switch rule

The base b logarithm of c is 1 divided by the base c logarithm of b.

logb(c) = 1 / logc(b)

For example:

log2(8) = 1 / log8(2)

Logarithm base change rule

The base b logarithm of x is base c logarithm of x divided by the base c logarithm of b.

logb(x) = logc(x) / logc(b)

For example, in order to calculate log2(8) in calculator, we need to change the base to 10:

log2(8) = log10(8) / log10(2)

See: log base change rule

Logarithm of negative number

The base b real logarithm of x when x<=0 is undefined when x is negative or equal to zero:

logb(x) is undefined when x ≤ 0

See: log of negative number

Logarithm of 0

The base b logarithm of zero is undefined:

logb(0) is undefined

The limit of the base b logarithm of x, when x approaches zero, is minus infinity:

\lim_{x\to 0^+}\textup{log}_b(x)=-\infty

See: log of zero

Logarithm of 1

The base b logarithm of one is zero:

logb(1) = 0

For example, teh base two logarithm of one is zero:

log2(1) = 0

See: log of one

Logarithm of infinity

The limit of the base b logarithm of x, when x approaches infinity, is equal to infinity:

lim logb(x) = ∞, when x→∞

See: log of infinity

Logarithm of the base

The base b logarithm of b is one:

logb(b) = 1

For example, the base two logarithm of two is one:

log2(2) = 1

Logarithm derivative

When

f (x) = logb(x)

Then the derivative of f(x):

f ' (x) = 1 / ( x ln(b) )

See: log derivative

Logarithm integral

The integral of logarithm of x:

∫ logb(x) dx = x ? ( logb(x) - 1 / ln(b) ) + C

For examp

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