Here you will learn what are the properties of logarithms and fundamental identities of logarithm with examples.<\/p>\n
Let’s begin –<\/p>\n
Every positive real number N can be expressed in exponential form as \\(a^x\\) = N where ‘a’ is also a positive real number different than unity and is called the base and ‘x’ is called an exponent.<\/p>\n
We can write the relation \\(a^x\\) = N in logarithmic form as \\(log_aN\\) = x. Hence \\(a^x\\) = N <=> \\(log_aN\\) = x. Hence logarithm of a number to some base is the exponent by which the base must be raised in order to get that number.<\/p>\n
\\(log_aN\\) is defined only when<\/p>\n
\n(i) N > 0<\/p>\n
(ii) a > 0<\/p>\n
(iii) \\(a\\neq1\\)<\/p>\n<\/blockquote>\n
Properties of Logarithms<\/h2>\n
If m, n are arbitrary positive numbers where a>0,\\(a\\neq1\\) and x is any real number, then-<\/p>\n
\n(a)\u00a0 \\(log_a mn\\) = \\(log_a m\\) + \\(log_a n\\)<\/p>\n
(b)\u00a0 \\(log_a\\)\\(m\\over n\\) = \\(log_a m\\) – \\(log_a n\\)<\/p>\n
(c)\u00a0 \\(log_a\\)\\(m^x\\) = x\\(log_a m\\)<\/p>\n<\/blockquote>\n\n\n
Example : <\/span> If \\(a^2\\) + \\(b^2\\) = 23ab, then show that \\(log (a + b)\\over 5\\)= \\(1\\over 2\\)(log a + log b).<\/p>\n
Solution : <\/span>\\(a^2\\) + \\(b^2\\) = \\((a+b)^2\\) – 2ab = 23ab
=> \\((a+b)^2\\) = 25ab
=> a+b = 5\\(\\sqrt{ab}\\)
L.H.S. = \\(log(a+b)\\over 5\\) = \\(log(5 \\sqrt{ab}) \\over 5\\) = \\(1 \\over 2\\)log ab = \\(1 \\over 2\\)(log a + log b) = R.H.S.
<\/p>\n\n\nFundamental Identities<\/h2>\n
Using the basic definition of logarithm we have 2 important deductions:<\/p>\n
\n(a) \\(log_NN\\) = 1 i.e logarithm of a number to the same base is 1.<\/p>\n
(b) \\(log_N\\)\\(1\\over N\\) = -1 i.e logarithm of a number to the base as its reciprocal is -1.<\/p>\n<\/blockquote>\n
Note :<\/strong><\/p>\n
\nN = \\((a)^{\\log_a N}\\) e.g. \\(2^{\\log_2 7}\\) = 7<\/p>\n<\/blockquote>\n\n\n
Example : <\/span> If \\(log_4m\\) = 3,then find the value of m.<\/p>\n
Solution : <\/span>\\(log_4m=3\\) => \\(m=4^3\\) => \\(m=64\\).
<\/p>\n\n\nHope you learnt what are the properties of logarithms and fundamental identities of logarithm. To learn more practice more questions and get ahead in competition. Good Luck!<\/p>\n\n\n