#### Table of Contents

# 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. 2^{3}=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: Log_{2} 8 =3

**I) find the values for the following logarithms:**

- log
_{3}27 - log
_{10}100 - log
_{4}2 - log
_{10}(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 **2 ^{3}=8**, this is represented as

**log**

_{2}8=3This is how we interconvert exponents into logarithms and vice versa.

**II) convert the following logarithms into exponents**

- log
_{a}b = x - log
_{25}5= 1/2 - log
_{3}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^{loga x} = x

log_{a} 0 is undefined

**III) Solve the following logarithms:**

- log
_{5}x = log_{5}2 + log_{5}7 - log
_{b}x+log_{b}1/x - log
_{5}15 - log_{5}3 - log
_{5}125 - log
_{16}4

Solutions are at the end of the article.

Sometimes we drop bases in logarithms for base 10 and natural logarithms. **Log _{10}** 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**:
log_{3} 27 = log_{3} $3^3$

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

= 3log_{3} 3

Using the property log_{a} a=1

= 3

**2)** ans=2

**Sol:**

log_{10} 100 = log_{10} $10^2$

Using the property log_{b} $a^n$ = n log_{b} a

= 2log_{10} 10

Using the property log_{a} a=1

=2

**3)** ans=½

**Sol:**

log_{4} 2 = log_{22} 2

Using the property log_{bn} a = $\frac{1}{n}$ log_{b} a

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

Using the property log_{a} a=1

= $\frac{1}{2}$

**4)** ans=-1

**Sol:**

Log_{10} (0.1) = log_{10} $\frac{1}{10}$

Using the property Log_{a} $\frac{x}{y}$ =log_{a} x- log_{a} y

= - log_{10} 10

Using the property log_{a} a=1
=-1

**II)**

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

**III)**

**1)** x=14

**Sol:**

log_{5} x = log_{5} 2 + log_{5} 7

Using the property Log_{a} xy=log_{a} x+ log_{a} y

log_{5} x = log_{5} 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:**

log_{5} 15 - log_{5} 3

Using the property Log_{a} $\frac{x}{y}$ =log_{a} x- log_{a} y

=Log_{5} $\frac{15}{3}$

= log_{5} 5

Using the property Log_{a} 1 = 0

=0

**4)** ans =3

**Sol:**

log_{5} 125

=log_{5} $5^3$

Using the property log_{b} a^{n} = n log_{b} a

=3 log_{5} 5

Using the property log_{a} a=1

=3

**5)** ans =½

**Sol:**

log_{16} 4

=log_{42} 4

Using the property log_{bn} a = $\frac{1}{n}$ log_{b} a

=½ log_{4} 4

Using the property log_{a} a=1

=½

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