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.

We learned that 4^{2}=16 . We can also write this another way :

Log^{4}16 = 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.

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 log_{a}N = x.

Remember that logarithms are defined only for positive real numbers.

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

So, log_{a}N is also unique.

Exponential function Logarithmic function

a^{x} = N x = log_{a}N

b^{y} = N y = log_{b}N

x^{z} = Z y = log_{x}Z

Functions defined by such equations are called logarithmic functions.

We can express exponential forms in logarithmic form.

**Exponential form Logarithmic form **

2^{4} = 16 4 = log_{2}16

1/9 = 1/3^{2} –2 = log_{3}1/9

= 3^{-2}

If a^{x} = N1 (a ≠1, a > 0),

then x = log_{a}N

Observe the following examples:

2^{6}= 64 can be written as log_{2}64 = 6

4^{3} = 64 can be written as log_{4}64 = 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.

I Write the following in logarithmic forms.

- 2
^{4}= 16 - 10
^{4}= 10000 - 3
^{3}= 27 - 10
^{-1}= 0.01 - 6
^{3}= 216

- 2
^{4}= 16

log_{2}16 = 4 - 10
^{4}= 10000

log_{10}10000 = 4 - 3
^{3}= 27

log_{3}27 = 3 - 10
^{-1}= 0.01

log_{10}0.01 = – 1 - 6
^{3}= 216

log_{6}216 = 3 - log636 = 2
- log5125 = 3
- log100.1 = –1
- log4256 = 4
- log981 = 2

**II Express each of the following in exponential forms**

- log
_{6}36 = 2

6^{2}= 36 log

_{5}125 = 3

5^{3}= 125log

_{10}0.1 = –1

10^{-1}= 0.1log

_{4}256 = 4

4^{4}= 256- log
_{9}81 = 2

9^{2}= 81

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