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.
a^m = b is exponential form.
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_ab = m if and only if a^m = b
'log' being the abbreviation of the word 'logarithm.

Properties of logarithm
1) log_a1 = 0 because a⁰=1
2) log_aa = 1 because a¹=a
3) log_aa^x = x because a^x =a^x

Evaluating logarithm

1) log₃27
Solution : We know from definition that
log_ab = m, if and only if a^m = b, a > 0 and a≠1
Let m = log₃27
∴ 3^m = 27
3^m = 3³
∴ m = 3
so, log₃27= 3

2) log₂√(32)
Solution : We know from definition that
log_ab = m, if and only if a^m = b, a > 0 and a≠1
Let m = log₂√(32)
∴ 2^m = √(32)
2^m = (2⁵)^1/2
2^m = 2^5/2
∴ m = 5/2
so, log₂√(32)= 5/2

Exponential to logarithmic form

2⁷ = 128	⇒ log ₂128 = 7
3⁴ = 81	⇒ log ₃81 = 4
6⁰ = 1	⇒ log ₆1 = 0
a^-b = c	⇒ log _ac = -b
4³ = 64	⇒ log ₄ 64 = 3
7² = 49	⇒ log ₇ 49 = 2
5³ = 125	⇒ log ₅ 125 = 3
10³ = 1000	⇒ log ₁₀ 1000 = 3
8³ = 512	⇒ log ₈ 512 = 3
15^-2 = 1/225	⇒ log ₁₅ (1/225) = -2

Logarithmic to Exponential form

log₂ 1 = 0	⇒ 2⁰ = 1
log₅25 = 2	⇒ 5² = 25
log₇343 = 3	⇒ 7³ = 343
log₁₀100 = 2	⇒ 10² = 100
log₈16 = 4/3	⇒ 8^4/3 = 16

11th grade math

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