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. 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
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I) find the values for the following logarithms:
- log3 27
- log10 100
- log4 2
- 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
- loga b = x
- log25 5= 1/2
- 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:
- log5 x = log5 2 + log5 7
- logb x+logb 1/x
- log5 15 - log5 3
- log5 125
- 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)
- $a^x$
- 5= 25$\frac{1}{2}$
- 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
=½
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