Introduction to Logarithms |
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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. |
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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. |
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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. |
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Try these questions |
I. Write the following in logarithmic forms. |
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1. |
24 = 16 |
2. |
104 = 10000 |
3. |
33 = 27 |
4. |
10-1 = 0.01 |
5. |
63 = 216 |
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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 |
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II Express each of the following in exponential forms |
| 6. |
log636 = 2 |
7. |
log5125 = 3 |
8. |
log100.1 = –1 |
9. |
log4256 = 4 |
10. |
log981 = 2 |
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II. Answers to questions |
6. |
log636 = 2
62 = 36 |
7. |
log5125 = 3
53 = 125 |
8. |
log100.1 = –1
10-1 = 0.1 |
9. |
log4256 = 4
44 = 256 |
10. |
log981 = 2
92 = 81 |
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