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:

25 = 32

Then

log2(32) = 5

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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,

-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}

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

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:

log8(4 ∙ 2) = log8(4) + log8(2)

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:

log8(5 / 4) = log8(5) – log8(4)

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:

log8(35) = 5 ∙ log8(3)

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:

log3(4) = 1 / log4(3)

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:

log3(4) = log8(4) / log8(3)

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 ≤ 0

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^+}{log}_b(x)=-\infty

Logarithm of 1

The base b logarithm of one is zero:

logb(1) = 0

For example:

log4(1) = 0

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→∞

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:

log4(16) = 2

Logarithm derivative

When

f (x) = logb(x)

Then the derivative of f(x):

f ‘ (x) = 1 / ( x ln(b) )

Logarithm integral

The integral of logarithm of x:

∫ logb(x) dx = x ∙ ( logb(x) – 1 / ln(b) ) + C

For example:

∫ log4(x) dx = x ∙ ( log4(x) – 1 / ln(4) ) + C

Logarithm approximation

log2(x) ≈ n + (x/2n – 1) ,

Complex logarithm

For complex number z:

z = re = x + iy

The complex logarithm will be (n = …-2,-1,0,1,2,…):

log z = ln(r) + i(θ+2nπ) = ln(√(x2+y2)) + i·arctan(y/x))

Logarithm problems and answers

Problem #1

Find x for

log2(x) + log2(x-3) = 2

Solution:

Using the product rule:

log2(x∙(x-3)) = 2

Changing the logarithm form according to the logarithm definition:

∙ (x-3) = 22

Or

x– 3– 4 = 0

Solving the quadratic equation:

x1,2 = [3±√(9+16) ] / 2 = [3±5] / 2 = 4,-1

Since the logarithm is not defined for negative numbers, the answer is:

x = 4

Problem #2

Find x for

log3(x+2) – log3(x) = 2

Solution:

Using the quotient rule:

log3((x+2) / x) = 2

Changing the logarithm form according to the logarithm definition:

(x+2)/x = 32

Or

x+2 = 9x

Or

8x = 2

Or

x = 0.25

Graph of log(x)

log(x) is not defined for real non positive values of x:

Logarithms table

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