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We all have used a log table to solve numerous mathematics questions. But, what are exactly logs? At its core, logarithms (logs) are another way to express exponents. For example, we know that 35 is 243. Now, if I asked you what is the number that you place is instead of the question mark 3? you will say it is 5. But, what if I asked you to replace the question mark with a number in 3? =53 Is there any number such that 3 raised to it gives 53? No, right? So, then how to solve this question. The answer is a log.
What are Logs (Logarithms)?
Logs (or) logarithms is the inverse function to exponentiation. For example, we know that 2 raised to the 2nd power equals 4. We can express this in the exponential form like this: 23=8.
23=8 log28=3
You can express the same thing in the logarithmic equation like this: log28=3. Here, both the equations are expressing the same relationships between the numbers 2,3 and 8. The key difference is that in the exponential form power 8 is isolated. While in the logarithmic form the exponent 3 is isolated. In simple words, logarithm takes into consideration the number of occurrences of the same factor in repeated multiplication. For example, 1000= 10 × 10 × 10=103; we can express the same things in logarithmic form in this way: log101000=3.
Now, let’s understand how you can read a logarithmic equation.
- bx=a logba=x
As you can guess here log stands for logarithm. You need to read the right side of the arrow-like this: ‘Logarithm of a to the base b is equal to x’.
Here,
- a and b are two positive real numbers
- a is called an argument while b is called the base
- x is the real number
You can understand two things from the symbol:
- bx=a logba=x: This is called ‘exponential to log form’.
- logba=x bx=a: This is called ‘log to exponential form’.
Read the table given below properly to understand the conversions from exponential form to the logarithmic form:
Exponential Form | Logarithmic Form |
23=8 | log2(8)=3 |
44=256 | log4(256)=4 |
52=25 | log5(25)=2 |
e2= 7.389 | loge(7.389)=2 |
103=1000 | log10(1000)=3 |
Logs are further categorized into two types: Natural and Common Log. Let’s understand them in great detail.
Natural Log:
Logs that have a base e are called natural logs. The constant ‘e’ is the Napier constant. The value e is approximately 2.718281828. Natural logs are commonly used as calculus in pure mathematics. They have the same properties as all the other logs. Natural logs are usually represented as In i.e logₑ = ln.
For example, ex = 2 ⇒ logₑ 2 = x (or) ln 2 = x.
An example of the natural log is as follows:
102=100 log10100=2 or log 100=2
Common Log:
Common logs have a fixed base of 10. They are expressed as log10. Even if you just write a log it is understood that the base is 10. They are also commonly known as decadic logarithms and decimal logarithms. They have a wide application in science and engineering.
Rules of Logs:
Let’s understand some common rules of logs that will help you to solve any logarithmic equation:
Product Rule:
This is one of the most important and commonly used rules. So, understand this carefully.
- logamn=logam+logan
All the bases of the log here are the same. This is originally derived from the product rule of exponents:
- xm. xn=xm+n.
Example:
- log 8=log (4×2)=log4+log2
Quotient Rule of Log:
This is another very important rule. The rule states that the quotients of two numbers will always be the difference between the logarithms of the individual numbers.
- logamn=logam-logan
In this rule as well, all the bases of the log here are the same. It was derived from the quotient rule of exponents: xm/ xn=xm-n.
Example:
- log4=log(8/2)=log8-log2
Log 1:
The value of log 1 will always be 0 irrespective of the base. This is because one of the properties of exponents states that the a0=1, for any a. When we convert this into a log form we get loga1=0, for any a. If we apply this in the natural log we get, e0=1.
Logaa:
The rule states that the logarithm of any number to the same base is always 1. As we all know that a1=a. Converting this into log form we get logaa=1.
Example:
- log55=1
- log33=1
- In e=1
Conclusion:
As you can see, the log is not at all difficult to understand. If you have understood exponents then solving logarithmic equations will be easy for you. Even the major rules of logarithms are very easy to remember. Just remember that log is the inverse function to exponentiation and counts the number of occurrences of the same factor in repeated multiplication.
