Logarithms Learning

Logarithms function is the inverse of exponential function. If f(x)=logb(x) where b is the base of the logarithm function ,b>0and b is not equal to one .The power to which is base ten and base e(e=2.5466) obtain raised in number. The logarithm of a power equals the exponent multiplied with the logarithm of the base.

Examples:

1. f(x) = log2x , base is 2

2. g(x) = log4x, base is 4

3. h(x) = log0.5x, base is 5


Properties of logarithms Learning:


1.logbmn  =  logbm + logbn

The logarithm of a product is equal to the sum of the logarithms of each factor."

2. logb m/n  = logbm − logbn

The logarithm of a quotient is equal to logarithm of the numerator minus the logarithm of the denominator.

3. logb yn = n logby

The logarithm of a power of y equal to exponent of that power times the logarithm of y.

Some basic property of logarithms

loga1=0

logaa=1

logaxa=alogax


Change of base Rule for logarithms Learning:


By the definition of the base two logarithm

log2x = log2x

=> x = 2log2x ( as log2x = a then x = 2a)

(since immediate consequences of the definition of logarithms ) that the logarithm of a power equals the exponent multiplied with the logarithm of the base. Therefore by taking natural logarithm on both sides of the proceeding equation, obtain

ln(x) = log2(x)ln(2)

(since, immediate consequences of the definition of logarithms ) that the logarithm of a power equals the exponent multiplied with the logarithm of the base. Therefore by taking natural logarithm on both sides of the preceding equation obtains

Solving for base two logarithm gives the same formula as before:

log2(x)=`(ln(x))/(ln(2))`


Examples of logarithms Learning:


Example1:Solve log23+log28=log2 (4x)

Solution: logarithmic function log23+log28=log2 (4x)

log2(3*8)  = log2(4x)

log2(4x)=log2(24) by property of logarithm (1)

Equate both sides, s the bases are same 2,  4x = 24

Simplification: x = 24/4

Answer = 6

Example2: log654-log69

Solution:

lo654-log69=log6(54/9) by property of logarithm (2)

simplification: log66

Answer = 1

Examble3: logarithm Solving equation in    log284

Solution:Converting the logarithmic equation

=  4log28  by property logarithm (3)

Simplification: 4log223 = 12log22

Answer=12