## 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

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

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