Logarithm of a number to a given base is the index or the power to which the base must be raised to produce the number. That is, to make it equal to the given number.
Let us consider the three quantities, sat a, x and n and they can be related as follows.
If a^{x} = n, where n > 0, a > 0 and a ≠ 1, then x is said to be the logarithm of the number n to the base 'a', symbolically it can be expressed as follows.
log_{a}n = x
Here, 'n' is called argument.
Law 1 :
Logarithm of product of two numbers is equal to the sum of the logarithms of the numbers to the same base.
log_{a}mn = log_{a}m + log_{a}n
Law 2 :
Logarithm of the quotient of two numbers is equal to the difference of their logarithms to the same base.
log_{a}(m/n) = log_{a}m - log_{a}n
Law 3 :
Logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number to the same base.
log_{a}m^{n} = nlog_{a}m
Example 1 :
If log_{a}m = x, then
m = a^{x}
Example 2 :
If k = x^{y}, then
log_{x}k = y
Multiplication of two or more logarithms can be simplified, if one of the following two conditions is met.
Condition (i) :
The argument of the first logarithm and the base of the second logarithm must be same.
Condition (ii) :
The base of the first logarithm and the argument of the second logarithm must be same.
Let us see how the multiplication of two logarithms can be simplified in the following examples.
Example 1 :
Simplify : log_{a}b ⋅ log_{b}c
In the above two logarithms, the argument of the first logarithm and the base of the second logarithm are same.
So, we can simplify the multiplication of above two logarithms by removing the part circled in red color.
Example 2 :
Simplify : log_{x}y ⋅ log_{z}x
In the above two logarithms, the base of the first logarithm and the argument of the second logarithm are same.
So, we can simplify the multiplication of above two logarithms by removing the part marked in red color.
If the logarithm of a number to any base is given, then the logarithm of the same number to any other base can be determined from the following relation.
log_{b}a = log_{x}a ⋅ log_{b}x
log_{b}a = log_{x}a / log_{x}b
More clearly,
In the above example, the base of the given logarithm 'b' is changed to 'x'.
In this way, we can change the base of the given logarithm to any other base.
Property 1 :
If you switch a logarithm from numerator to denominator or denominator to numerator, we have to interchange the argument and base.
Examples :
log_{a}b = 1/log_{b}a
5/(2log_{x}y) = 5log_{y}x/2
Property 2 :
Logarithm of any number to the same base is equal to 1.
Examples :
log_{x}x = 1
log_{3}3 = 1
The above property will not work, if both the base and argument are 1.
log_{1}1 ≠ 1
The reason for the above result is explained in the next property.
Property 3 :
Logarithm of 1 to any base is equal to 0.
Examples :
log_{x}1 = 0
log_{3}1 = 0
log_{1}1 = 0
1. If the base is not given in a logarithm, it can be taken as 10.
log10 = log_{10}10 = 1
log1 = log_{10}1 = 0
2. Logarithm using base 10 is called Common logarithm and logarithm using base 'e' is called Natural logarithm.
Here, e ≈ 2.718 called exponential number.
Problem 1 :
Find the logarithm of 125 to the base 5.
Solution :
Write 125 as a power of 5.
125 = 5 x 5 x 5
= 5^{3}
log_{5}125 = log_{5}(5)^{3}
= 3log_{5}5
= 3(1)
= 3
Problem 2 :
Find the logarithm 1728 to the base 2√3.
Solution :
Write 1728 as a power of 2√3.
1728 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3
1728 = 2^{6} x 3^{3}
1728 = 2^{6} x [(√3)^{2}]^{3}
1728 = 2^{6} x (√3)^{6}
1728 = (2√3)^{6}
log_{2}_{√3}(1728) = log_{2}_{√3}(2√3)^{6}
Using the power rule of logarithms,
= 6log_{2}_{√3}(2√3)
= 6(1)
= 6
Problem 3 :
Find the logarithm of 0.001 to the base 0.1.
Solution :
log_{0.1}0.001 = log_{0.1}(0.1)^{3}
= 3log_{0.1}0.1
= 3(1)
= 3
Problem 4 :
Find the logarithm 1/64 to the base 4.
Solution :
log_{4}(1/64) = log_{4}1 - log_{4}64
= 0 - log_{4}(4)^{3}
= -3log_{4}4
= -3(1)
= -3
Problem 5 :
Find the logarithm of 0.3333...... to the base 3.
Solution :
log_{3}(0.3333......) = log_{3}(1/3)
= log_{3}1 - log_{3}3
= 0 - 1
= -1
Problem 6 :
If log_{y}(√2) = 1/4, find the value of y.
Write the equation in exponential form.
√2 = y^{1/4}
Raise to the power 4 on both sides.
(√2)^{4} = (y^{1/4})^{4}
(2^{1/2})^{4} = y
2^{2} = y
4 = y
Problem 7 :
Simplify :
(1/2)log_{10}25 - 2log_{10}3 + log_{10}18
Solution :
= (1/2)log_{10}25 - 2log_{10}3 + log_{10}18
Using power rule of logarithms,
= log_{10}25^{1/2} - log_{10}3^{2} + log_{10}18
= log_{10}(5^{2})^{1/2} - log_{10}3^{2} + log_{10}18
= log_{10}5 - log_{10}9 + log_{10}18
= log_{10}5 + log_{10}18 - log_{10}9
Using the product rule of logarithms,
= log_{10}(5 x 18) - log_{10}9
= log_{10}90 - log_{10}9
Using the quotient rule of logarithms,
= log_{10}(90/9)
= log_{10}10
= 1
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