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LOGARITHMS

Shubham Jaiswal

22/04/2017 0 0

Definition and Basics of Logarithms--Part II

Saying that log_aM =x means exactly the same thing as saying a^x = M .

In other words:

log_aM is 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 2³the answer is "3." So log₂8=3.

Here's another way that remembering the rule:

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

can make some things almost obvious. For example, what is 2 ^log2⁵? Note that ^log₂⁵is the power to which 2 is being raised.

But ^log₂⁵ 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 ^log2⁵= 5.

Let's use

"log_aM is 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=2³and 32=2⁵so 8*32=2³2⁵ = 2³⁺⁵=2⁸.

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: log_a a^x = x and a^(loga ^x) = x

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

3. Quotient:

4. Power: log_a (x^p) = p log_a x

5. Change of base formula:

Careful!!

log_a (x + y) ≠ log_a x + log_a y

log_a (x – y) ≠ log_a x – log_a 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:

log_b(x) = y

For example when:

2⁴ = 16

Then

log₂(16) = 4

Logarithm as inverse function of exponential function

The logarithmic function,

y = log_b(x)

is the inverse function of the exponential function,

x = b^y

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

f (f ^-1(x)) = b^logb^(x) = x

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

f ^-1(f (x)) = log_b(b^x) = x

Natural logarithm (ln)

Natural logarithm is a logarithm to the base e:

ln(x) = log_e(x)

When e constant is the number:

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

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

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) = log_b(x)

Logarithm rules

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

log_b(x ? y) = log_b(x) + log_b(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.

log_b(x / y) = log_b(x) - log_b(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.

log_b(x ^y) = y ? log_b(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.

log_b(c) = 1 / log_c(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.

log_b(x) = log_c(x) / log_c(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:

log_b(x) is undefined when x ≤ 0

See: log of negative number

Logarithm of 0

The base b logarithm of zero is undefined:

log_b(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:

log_b(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 log_b(x) = ∞, when x→∞

See: log of infinity

Logarithm of the base

The base b logarithm of b is one:

log_b(b) = 1

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

log₂(2) = 1

Logarithm derivative

When

f (x) = log_b(x)

Then the derivative of f(x):

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

See: log derivative

Logarithm integral

The integral of logarithm of x:

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

For examp

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