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Algebra

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:

  • am × an = am + n
  • am ÷ an = am – n           (a ≠ 0)
  • (am)n = amn
  • (ab)= am × bm
  • (a ÷ b)= am ÷ bm    (b ≠ 0)
  • a-m  = 1/am                   (a ≠ 0)
  • a0 = 1

Definition of Logarithm:

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

Logarithmic form

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

Conversion from the Logarithmic Form into the Exponential Form:

Logarithmic FormExponential Form
log100.0001 = – 410-4 = 0.0001
log2128 = 727 = 128
log816 = 4/384/3 = 16
log96561 = 494 = 6561
log(1/16)(1/8) = 3/4(1/16)3/4 = 1/8
log2(1/4) = -294 = 6561
log0.50.125 = 30.53 = 0.125
logqp = rqr = p
log232 = 525 = 32
log(1/2)(1/8) = 3(1/2)3 = 1/8
log279 = 2/3272/3 = 9
log7343 = 373 = 243
log33 = 131 = 3
log6(1/36) = -26-2 = 1/36
log81(1/3) = – 1/481– 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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