Logarithms Intro

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, a 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

We learned that 42=16 . We can also write this 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 raise x to, to get y. Thus, the logarithm of a number to a given base is the index to which the base should be raised to get the given number.

Definition

If N and a, a ≠1 are any two positive real numbers and for some real x, if ax = N then x is said to be the logarithm of N to the base “a’, and is written as logaN = x.

Remember that logarithms are defined only for positive real numbers.

Also, there exists a unique x that satisfies the equation ax = N.

So, logaN is also unique.

       Exponential function       Logarithmic function

                ax = N                         x = logaN

                by = N                         y = logbN

                xz = Z                         y = logxZ

Functions defined by such equations are called logarithmic functions.

We can express exponential forms in logarithmic form.

     Exponential form              Logarithmic form

             24 = 16                         4 = log216

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

                 = 3-2              

If ax = N1 (a ≠1, a > 0),

then x = logaN

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.

Try these questions

I     Write the following in logarithmic forms.

  1. 24 = 16
  2. 104 = 10000
  3. 33 = 27
  4. 10-1 = 0.01
  5. 63 = 216

Answers to questions

  1. 24 = 16
    log216 = 4

  2. 104= 10000
    log1010000 = 4

  3. 33 = 27
    log327 = 3

  4. 10-1 = 0.01
    log100.01 = – 1

  5. 63 = 216
    log6216 = 3

  6. II     Express each of the following in exponential forms

  7. log636 = 2
  8. log5125 = 3
  9. log100.1 = –1
  10. log4256 = 4
  11. log981 = 2

II.     Answers to questions

  1. log636 = 2
    62 = 36

  2.  log5125 = 3
    53 = 125


  3. log100.1 = –1
    10-1 = 0.1


  4. log4256 = 4
    44 = 256


  5. log981 = 2
    92 = 81