Evaluating Logarithm
In this section, student will learn evaluating logarithm.The word logarithm was coined from two Greek words"logos" which means a ratio and "arithmos" means numbers.
Logarithms and exponents are closely related to each other. The inverse of these operations gives us two distinct operations viz. extracting roots and taking logarithms.
It is read as mth power of 'a' is 'b'.
For each positive real number a, a≠ 1, the unique real number 'm' is called the logarithm of 'b' to the base 'a' or
'log' being the abbreviation of the word 'logarithm.
Properties of logarithm
1) log _{a} 1 = 0 because a ^{0} =1
2) log _{a} a = 1 because a ^{1} =a
3) log _{a} a ^{x} = x because a ^{x} =a ^{x}
Evaluating logarithm
1) log _{3} 27Solution : We know from definition that
log _{a} b = m, if and only if a ^{m} = b, a > 0 and a≠1
Let m = log _{3} 27
∴ 3 ^{m} = 27
3 ^{m} = 3 ^{3}
∴ m = 3
so, log _{3} 27= 3
2) log _{2} √(32)
Solution : We know from definition that
log _{a} b = m, if and only if a ^{m} = b, a > 0 and a≠1
Let m = log _{2} √(32)
∴ 2 ^{m} = √(32)
2 ^{m} = (2 ^{5} ) ^{1/2}
2 ^{m} = 2 ^{5/2}
∴ m = 5/2
so, log _{2} √(32)= 5/2
Exponential to logarithmic form
2^{7} = 128 | ⇒ log _{2}128 = 7 |
3^{4} = 81 | ⇒ log _{3}81 = 4 |
6^{0} = 1 | ⇒ log _{6}1 = 0 |
a^{-b} = c | ⇒ log _{a}c = -b |
4^{3} = 64 | ⇒ log _{4} 64 = 3 |
7^{2} = 49 | ⇒ log _{7} 49 = 2 |
5^{3} = 125 | ⇒ log _{5} 125 = 3 |
10^{3} = 1000 | ⇒ log _{10} 1000 = 3 |
8^{3} = 512 | ⇒ log _{8} 512 = 3 |
15^{-2} = 1/225 | ⇒ log _{15} (1/225) = -2 |
log_{2} 1 = 0 | ⇒ 2^{0} = 1 |
log_{5}25 = 2 | ⇒ 5^{2} = 25 |
log_{7}343 = 3 | ⇒ 7^{3} = 343 |
log_{10}100 = 2 | ⇒ 10^{2} = 100 |
log_{8}16 = 4/3 | ⇒ 8^{4/3} = 16 |
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