A logarithm is a word and concept coined by John Napier, a Scottish mathematician. A logarithm is derived from the combination of two Greek words that are logos that means principle or thought and arithmos means a number.
A logarithm is the power to which must be raised to get a certain number. It is denoted by the log of a number. Example: log(x).
if y=ax
then, logay= x
a is the base.
x is the exponent.
where, a>0, a≠1, y≠0
For example: 25= 32
Log232= 5
Logarithmic Form | Exponential Form |
Log3 9= 2 | 32 = 9 |
Log3 81= 4 | 34 = 81 |
Log5 625= 4 | 54 = 625 |
The logarithm whose base is 10 is called the Common logarithm. It is basically how many 10 must be multiplied to get a given number. It is denoted as log 10 or log.
For example:
log 100
= log 10²
= 2log 10
= 2
The logarithm whose base is considered e (Euler's constant) is called Natural logarithm. It is the number of e that must be multiplied to get the given number.
For Example:
e= 2.71828
The natural log of a number for e.g. 56 is denoted by Ln56.
In 56= 4.02535169
1. logb (mn) = n logb m (Log of power rule)
Example: log10 (23)= 3log102
2. logb (mn)= logb m + logbn (Product rule)
Example: log10 (14)= log10 (7*2)= log10 7 + logb 2
3. logb (m/n)= logb m – logb n (Quotient rule)
Example: log10 (4/5)= logb 4 – logb 5
4. logb 1= 0
5. logb (m) = ln m / ln b (Change of base rule)
or logb (n) = log10n / log10 b
For example: log10 (7) = ln 7 / ln 10
6. ln(1/x) = -ln (x) (Log of reciprocal rule)
For example: . ln(1/5) = -ln (5)
logb (m+n) ≠ logb m + logb n
logb (m-n) ≠ logb m - logb n
In this technological era, people are always finding ways to do things in simpler and easier ways. Therefore, people invented calculators and logarithms to make mathematical equations easier to get solved.
So, the advantages of understanding the concept of Logarithm:
In many scientific research and studies, a logarithm is used.
Logarithms help to find the pH value in chemistry because the value for pH can be small, so we use the logarithm to have a range for using it for small numbers.
Logarithms are widely used in banking.
Logarithms are used to find the half-life of radioactive material.
It is used to find out the seismic waves.
It plays a very crucial role in the field of medicine or engineering.
1. If loga p= q, Express aq-1 in Terms of a and p.
Solution:
loga p= q
ap= q
ap/a = q/a
aq-1 = p/a
2. Solve: (3+log7 x)/(4 – 2 log7x) = 2
Solution:
3+log x = 8 – 4log7x
5log7x = 5
log7x = 1
x = 71 = 7
3. Solve (log 2x) 2 – log2x2 - 32 = 0. Given x is an Integer.
Solution:
(log2x)2 – log 2x4 - 32 = 0.
⇒ (log 2x)2 – 4log2 x - 32 = 0......(1)
log 2x = y (say)
(i) ⇒ y2 – 4y – 32 = 0
⇒ y2 – 8y + 4y – 32 = 0
⇒ y (y – 8) + 4 (y – 8) = 0
⇒ (y – 8) (y + 4) = 0
⇒ y = 8, -4
⇒ log2x = 8 or log2x = - 4
X=28 = 256.
Since, x is an integer therefore, x = 256.
4. Express [\[\frac{1}{3}\]]\[^{4}\] = \[\frac{1}{81}\] in Logarithmic Form.
Answer:
Take the log of base \[\frac{1}{3}\] on both sides
[\[\frac{1}{3}\]]\[^{4}\] = \[\frac{1}{81}\]
⇒ log\[_{\frac{1}{3}}\] [\[\frac{1}{3}\]]\[^{4}\] = log\[_{\frac{1}{3}}\] \[\frac{1}{81}\]
⇒ 4log\[_{\frac{1}{3}}\] \[\frac{1}{3}\] = log\[_{\frac{1}{3}}\] \[\frac{1}{81}\] (Since log\[_{b}\] a\[^{n}\] = n log\[_{b}\] a)
⇒ 4 = log\[_{\frac{1}{3}}\] \[\frac{1}{81}\] (Since log\[_{b}\] b = 1)
⇒ log\[_{\frac{1}{3}}\] \[\frac{1}{81}\] = 4
Hence, the logarithmic form is,
log\[_{\frac{1}{3}}\] \[\frac{1}{81}\] = 4 |
1. Why is it Necessary to Know the Properties of the Logarithm?
Ans: While studying, you will witness many properties like commutative, associative, and distributive properties. These types of properties are introduced to help us to solve our large equations or questions with much ease.
The main purpose of the logarithm is to solve complex and time-consuming sums with not only maximum accuracy but also in a much-simplified way. The properties of the product rule, quotient rule, and log of power rule are used to solve the difficult and large sums.
Hence, we need to know and learn the properties of the logarithm to solve the difficult equations in the easiest way possible.
2. What is the Value of log1 and log0?
Ans: log 0 = undefined
The logarithm can only be applied to numbers that are greater than 0. Log 0 is undefined since you cannot get 0 by raising anything to the power of anything else. With only infinitely large and negative power, we can approach zero otherwise, it is impossible.
Log₂1 = 0, Log₃1 = 0, Log₁₀1 = 0
Log1= 0
No matter what the base of the algorithm, we find that log 1 is always 0. The reason for this is that any number that is raised to the power of 0 is always equal to 1.
Therefore, ln1 is also equal to 0.
3. What is e? How is it Important?
Ans: The number, e is Euler’s number. e is a very important mathematical constant. The value of e is equal to 2.71828 approximately. e can be used as a base in logarithm, which is called the natural logarithm. It is denoted by ln x.
Note that ln(e)=1 and ln(1)=0.
The number e is very important in mathematics. Euler’s identity also states a very significant identity i.e; e^iπ+1=0. e is an irrational number and it is not a root of any non-zero polynomial having any rational coefficients.
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