This section introduces you to logarithms. It starts with defining the term logarithms and explains its importance. It covers the properties of logarithms and then goes on to explain the simple laws of logarithms. It also covers in detail the change of base of a logarithm.

Section 2: Objectives

By the end of the section, you will be able to:

1. Define a logarithm and its properties.
2. Explain the simple laws of logarithms.
3. Explain the theorem of change of base of a logarithm.

Topic 2.1  Introduction

     As you know multiplication is a shortcut for addition,
      for example, 5 3 = 5 + 5 + 5.
     
      Exponents are a shortcut for multiplication, for example

     5³ = 5 5 5. Likewise Logarithm is a shortcut for exponents.

     In this section you will learn some simple laws of logarithms.

     Logarithms are very useful in such calculations.

     They make even difficult calculations quite easy.

Defining Logarithms

     You know that 4² = 16, right? Well, we can write it in another way :

     Log₄16 = 2

     This is a log with subscript of 4. The equation is read as
     "the log to the base 4 of 16 is 2".

     The log to the base x of y is the number you can raise x to get y.

     log is the exponent.

     log of some number is the exponent you have to raise the base to get      that number.

Definition:
     
      If N and a, a1 are any two positive real numbers and for some
      real x if a^x  = N then x is said to be the logarithm of N to the
      base ‘a’, it is written as log_aN = x.

     Remember that logarithms are defined only for positive real numbers.

     Also that there exists a unique x which satisfies the equation a^x = N.

     so log_aN is also unique.

     Exponential function logarithmic function

     a^x = N                     x = log_aN

     b^y = N                     y = log_bN

     x^y = Z                     y = log_xZ

     Functions defined by such equations are called logarithmic functions.

     We can express exponential forms as logarithmic forms.

     Exponential form             logarithmic form

     2⁴ = 16                           4 = log₂16

     1/9 = 1/3² = 3^–2             –2 = log₃1/9


     If a^x = N₁ (a1, a > 0)

      Then x = log_aN

     Observe the following examples:

     2⁶ = 64 can be written as log₂64 = 6

     4³ = 64 can be written as log₄64 = 3

     From these examples, we know that logarithms of the same number
      i.e. 64 with two different bases i.e. 2 and 4 are different.

     Therefore, The logarithms of the same number to different bases are      different.

Properties of Logarithms

Property 1 : log_a1 = 0

     Recall that if a0, a⁰ = 1 therefore, log_a1 = 0

     hence the logarithm of 1 to any base is 0.

     Example

     log₂1 = 0; because in the exponential equation we know that 2⁰ = 1

      log_1/21 = 0; because in the exponential equation

     we know that (½)⁰ = 1

The logarithm of any unity to any non–zero base is zero.

Property 2 : log_aa = 1

     Recall that in exponents a¹ = a log_aa =1 (where a0, a > 0)

     Example

     log₁₀10 = 1 because in the exponential equation
      we know that 10¹ = 10

     log₇7 = 1 because in the exponential equation we know that 7¹ = 7

     log_ee = 1 because in the exponential equation we know that e¹ = e

The logarithm of any non–zero positive number to the same
base is unity.

Property 3 : log_aa^x = x

     Recall that a^x = a^x

     Example

     Since you know that 3⁴ = 3⁴, you can write the logarithm equation as

      log₃3⁴ = 4

Logarithmic Functions

     Let ‘a’ be a positive real number and a1. The function f: (0,)R
      is defined by

     f(x) = log_ax , x (0,) is called a logarithmic function.
     
      If log_ax = log_ay x = y

Natural Logarithms

     The logarithms which are computed to the base e = 2.718 . . . are      called natural logarithms (Naperian). It can be written as log_ex (l_nx)

Common Logarithms

     The logarithms that are computed to the base 10 are called common      (Briggs) logarithms and can be written as log₁₀x.

     (i) The domain of the logarithmic function = set of positive
         real numbers (0,)

     Range = set of real numbers (–,).

Logarithmic Symbols

1. If (a > 1, n >1) or (0 < n < 1, 0 < a < 1) then log_an > 0
   
   
2. If (n > 1, 0 < a < 1) or (0 < n < 1, a > 1) then log_an < 0