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 log_b 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:

a^b=x ------> b= log_a 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. log_a 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: Log_a xy = log_a x+ log_a y

Quotient rule: Log_a $\frac{x}{y}$ = log_a x- log_a y

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

Power rule: (ii) log_bⁿ a= $\frac{1}{n}$ log_b a

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

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

log_a 1 = 0

Other properties:

log_a a= 1

If log_b a= log_b c then a=c

log_b $\frac{1}{a}$=log_{$\frac{1}{b}$} a= -log_a a

a^{log_a x} = x

log_a 0 is undefined

III) Solve the following logarithms:

1. log₅ x = log₅ 2 + log₅ 7
2. log_b x+log_b 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 (=log_e 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 log_a a=1

= 3

2) ans=2

Sol:

log₁₀ 100 = log₁₀ $10^2$

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

= 2log₁₀ 10

Using the property log_a a=1

=2

3) ans=½

Sol:

log₄ 2 = log_2² 2

Using the property log_bⁿ a = $\frac{1}{n}$ log_b a

= $\frac{1}{2}$ log₂ 2

Using the property log_a a=1

= $\frac{1}{2}$

4) ans=-1

Sol:

Log₁₀ (0.1) = log₁₀ $\frac{1}{10}$

Using the property Log_a $\frac{x}{y}$ =log_a x- log_a y

= - log₁₀ 10

Using the property log_a 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 Log_a xy=log_a x+ log_a y

log₅ x = log₅ 2*7

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

x=14

2) ans=0

Sol:

log_b x+log_b 1/x

Using the property Log_a xy =log_a x+ log_a y

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

=log_b 1

Using the property Log_a 1 = 0

=0

3) ans= 0

Sol:

log₅ 15 - log₅ 3

Using the property Log_a $\frac{x}{y}$ =log_a x- log_a y

=Log₅ $\frac{15}{3}$

= log₅ 5

Using the property Log_a 1 = 0

=0

4) ans =3

Sol:

log₅ 125

=log₅ $5^3$

Using the property log_b aⁿ = n log_b a

=3 log₅ 5

Using the property log_a a=1

=3

5) ans =½

Sol:

log₁₆ 4

=log_4² 4

Using the property log_bⁿ a = $\frac{1}{n}$ log_b a

=½ log₄ 4

Using the property log_a a=1

=½