Concept of Logarithm

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In this article, we shall study the concept of logarithms and interconversion between exponential form and logarithmic form.

Laws of Indices:

• a^m × aⁿ = a^{m + n}
• a^m ÷ aⁿ = a^{m – n}           (a ≠ 0)
• (a^m)ⁿ = a^mn
• (ab)^m = a^m × b^m
• (a ÷ b)^m = a^m ÷ b^m    (b ≠ 0)
• a^-m  = 1/a^m                   (a ≠ 0)
• a⁰ = 1

Definition of Logarithm:

If m = a^x, where a > 0 and m > 0 then x is called the logarithm of m to the base a and is written as x = log_am and read as a log of m to the base a where m, a, x are real numbers.

Notes:

• The logarithm of a negative number and zero are not defined.
• Logarithm to the base 10 are called common or Briggsian logarithms.
• Logarithms to the base e, where e is an irrational number whose value is e = 2.7182… are called natural or Naperian logarithms.

Conversion from the Exponential Form into Logarithmic Form:

Exponential Form	Logarithmic Form
ab = c	logac = b
52 = 25	log525 = 2
91/2 = 3	log93 = 1/2
(27)1/3 = 3	log273 = 1/3
10-3 = 1/1000	log10(1/1000) = -3
70 = 1	log71 = 0
83 = 512	log8512 = 3
323/5 = 8	log328 = 3/5
7-2 = 1/49	log7(1/49) = -2
10-2 = 1/100	log10(1/100) = -2
25 = 32	log232 = 5
9-1/2 = 1/3	log9(1/3) = -1/2
23 = 8	log28 = 3
10-1 = 0.1	log10(0.1) = -1
4-2 = 1/16	log4(1/16) = -2
80 = 1	log81 =0
53 = 125	log5125 = 3
27-1/3 = 1/9	log27(1/9) = -1/3
95/2 = 243	log9243 = 5/2
4-3 = 1/64	log4(1/64) = -3

Conversion from the Logarithmic Form into the Exponential Form:

Logarithmic Form	Exponential Form
log100.0001 = – 4	10-4 = 0.0001
log2128 = 7	27 = 128
log816 = 4/3	84/3 = 16
log96561 = 4	94 = 6561
log(1/16)(1/8) = 3/4	(1/16)3/4 = 1/8
log2(1/4) = -2	94 = 6561
log0.50.125 = 3	0.53 = 0.125
logqp = r	qr = p
log232 = 5	25 = 32
log(1/2)(1/8) = 3	(1/2)3 = 1/8
log279 = 2/3	272/3 = 9
log7343 = 3	73 = 243
log33 = 1	31 = 3
log6(1/36) = -2	6-2 = 1/36
log81(1/3) = – 1/4	81– 1/4 = 1/3

In the next article, we shall study to solve the problems based on the definition of a logarithm.

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