# Definition and properties

Logarithm: For a number and base, the power the base should have so that it is equal to the given number is called a logarithm.For example, the logarithm of 8 with base 2 means what power should 2 have so that it is equal to 8. 23=8, the answer is 3.

logarithm is represented as logb a where a and b are non-zero positive numbers.

Where a is the number inside the logarithmic function and b is the base. And we read it as log a base b.

Symbolically the above example can be represented as: Log2 8 =3

I) find the values for the following logarithms:

1. log3 27
2. log10 100
3. log4 2
4. log10 (0.1)

Solutions are at the end of the article.

Logarithm in fact is an inverse of exponential function. Let’s see how logarithm and exponents are interconvertible:

ab=x ------> b= loga x

Look at the example 23=8, this is represented as log2 8=3

This is how we interconvert exponents into logarithms and vice versa.

II) convert the following logarithms into exponents

1. loga b = x
2. log25 5= 1/2
3. log3 9= 2

Solutions are at the end of the article.

Properties:

For a, b, x, y being postive non-zero numbers.

Product rule: Loga xy = loga x+ loga y

Quotient rule: Loga $\frac{x}{y}$ = loga x- loga y

Power rule: (i) logb $a^m$ = m logb a

Power rule: (ii) logbn a= $\frac{1}{n}$ logb a

logb a= $\frac{1}{log_a b}$

logb a= $\frac{log_x a}{log_x b}$

loga 1 = 0

Other properties:

loga a= 1

If logb a= logb c then a=c

logb $\frac{1}{a}$=log$\frac{1}{b}$ a= -loga a

aloga x = x

loga 0 is undefined

III) Solve the following logarithms:

1. log5 x = log5 2 + log5 7
2. logb x+logb 1/x
3. log5 15 - log5 3
4. log5 125
5. log16 4

Solutions are at the end of the article.

Sometimes we drop bases in logarithms for base 10 and natural logarithms. Log10 a can be represented dropping 10 i.e. log a. And logarithm with natural base ’e’ can be represented as ln x (=loge x)

Solutions:

I

1) ans=3

Sol: log3 27 = log3 $3^3$

Using the property $log_b$ $a^n$ = n $log_b$ a

= 3log3 3

Using the property loga a=1

= 3

2) ans=2

Sol:

log10 100 = log10 $10^2$

Using the property logb $a^n$ = n logb a

= 2log10 10

Using the property loga a=1

=2

3) ans=½

Sol:

log4 2 = log22 2

Using the property logbn a = $\frac{1}{n}$ logb a

= $\frac{1}{2}$ log2 2

Using the property loga a=1

= $\frac{1}{2}$

4) ans=-1

Sol:

Log10 (0.1) = log10 $\frac{1}{10}$

Using the property Loga $\frac{x}{y}$ =loga x- loga y

= - log10 10

Using the property loga a=1 =-1

II)

1. $a^x$
2. 5= 25$\frac{1}{2}$
3. 9= $3^2$

III)

1) x=14

Sol:

log5 x = log5 2 + log5 7

Using the property Loga xy=loga x+ loga y

log5 x = log5 2*7

Using the property if loga b=loga c then b=c

x=14

2) ans=0

Sol:

logb x+logb 1/x

Using the property Loga xy =loga x+ loga y

=logb x* $\frac{1}{x}$

=logb 1

Using the property Loga 1 = 0

=0

3) ans= 0

Sol:

log5 15 - log5 3

Using the property Loga $\frac{x}{y}$ =loga x- loga y

=Log5 $\frac{15}{3}$

= log5 5

Using the property Loga 1 = 0

=0

4) ans =3

Sol:

log5 125

=log5 $5^3$

Using the property logb an = n logb a

=3 log5 5

Using the property loga a=1

=3

5) ans =½

Sol:

log16 4

=log42 4

Using the property logbn a = $\frac{1}{n}$ logb a

=½ log4 4

Using the property loga a=1