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Value of Log 0 is undefined. A Logarithm function is an inverse function to an exponential. A logarithm function is used to calculate the value of a variable and eliminate the exponential functions. The mathematical equation for a logarithm function can be expressed as logab = x, then ax = b. It is important to note that the variable “a” should always be a positive integer and it should not be equal to 1.
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Keyterms: Log, Logarithm, Exponential functions, Integer, Base, Logarithm functions, Natural logarithmic functions, Common logarithmic functions
Also Read: Types of Probability
What is a Logarithm Function?
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A Logarithm function is an inverse function to an exponential. A logarithm function is used to calculate the value of a variable and eliminate the exponential functions. The mathematical equation for a logarithm function can be expressed as,
If ax = b
Then,
logab = x
Where,
x → Log of a number
a → base of a logarithm function.
It is important to note that the variable “a” should always be a positive integer and it should not be equal to 1.
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Types of Logarithm Function?
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Logarithm functions are categorised into two types.
- Common logarithm function
- Natural logarithm function
Common logarithm function is the logarithm function with base 0 while natural logarithm function is the one with base e.
Derivation of log 0 value with base 10
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The log functions of 0 to the base 10 is expressed as
Log10 0
On the basis of the logarithm function,
Base = 10 and 10x = b
As we know, the logarithm function logab can only be defined if b > 0, and it is not possible to find the value of x if ax = 0.
Log10 0 = Not Defined
Thus, log0 10 or log of 0 is not defined.
Also Read: Permutations and Combinations
Derivation of log 0 value with base e
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The logarithmic function of 0 to the base e can be expressed as
loge 0
The representation of the natural log of 0 is Ln.
ln (0)
If ex = 0, no number can satisfy the equation when x is equal to any value.
Hence, log 0 is not defined.
Loge 0 = In (0) = Not defined
Logarithm Values Table
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Logarithm values from 1 to 10 to the base 10 are given below in a tabulated format.
| Log | Value |
|---|---|
| Log 1 | 0 |
| Log 2 | O.3010 |
| Log 3 | 0.4771 |
| Log 4 | 0.6020 |
| Log 5 | 0.6989 |
| Log 6 | 0.7781 |
| Log 7 | 0.8450 |
| Log 8 | 0.9030 |
| Log 9 | 0.9542 |
| Log 10 | 1 |
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Ln Values Table
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Logarithm values from 1 to 10 to the Base e are tabulated below.
| ln | Values |
|---|---|
| In (1) | 0 |
| In (2) | 0.693147 |
| In (3) | 1.098612 |
| In (4) | 1.386294 |
| In (5) | 1.609438 |
| In (6) | 1.791759 |
| In (7) | 1.94591 |
| In (8) | 2.079442 |
| In (9) | 2.197225 |
| In (10) | 2.302585 |
Also Read: Pascal’s Triangle
Things to Remember
- John Napier introduced the concept of logarithm in the 17th century
- The logarithm with base 10 is known as common or Briggsian, logarithms and can be written as log n. They are usually written without a base.
- The logarithm is the inverse process of exponentiation.
- The value of log 0 is not defined.
- Logarithmic functions are of two types. Natural logarithmic functions and common logarithmic functions.
- Common logarithm function is the logarithm function with base 0.
- Natural logarithm function is the one with base e.
Previous Year Questions
- The sum of the divisors of 24⋅33⋅53 is…..
- The value of =dxd[xnlogaxex]=…
- The value of dxd[xnlogaxex] is...[JKCET 2013]
- If a=log23,b=log25,c=log72, then log14063 in terms of a, b, ca,b,c is…...[BITSAT 2007]
- The general value of the real angle θ, which satisfies the equation,...[WBJEE 2019]
- The differential coefficient of f (Sinx) w.r.t. x where f(x)=logx is….[KCET 2004]
- The differential coefficient of log10x with respect to logx10 is….….[KCET 2016]
- If log72=?, then the value of log49(28) is….
- The derivative of y = xsinx is...[UPSEE 2016]
- The derivative of (logx)x with respect to log x is….
Sample Questions
Ques. Find the value of y such that logy 64 = 2. (3 Marks)
Ans. Given that, logy 64 = 2
According to the definition of the logarithm function,
if logab = x, then
ax = b ….(1)
a = y, b= 64, x = 2
Substitute the values in (1), we get
y2 = 64
Take square roots on both sides,
y = √64
Therefore, the value of y is 8.
Ques. Solve for y in log2 y = 6. (3 Marks)
Ans. The logarithm function of the above function can be written as 26 = y
Hence,
25 =2 x 2 x 2 x 2 x 2 x 2 =64
or
Y = 64
Ques. Find the value of x such that log x 81 = 2. (3 Marks)
Ans. Given that, log x 81 = 2
On the basis of Logarithm definition
If logx b=x
ax = b – (1)
a=x, b= 81, x =2
Substituting the value in equation (1), we get
x2 =81
Taking square root on both sides we get,
x = 9
Therefore, the value of x = 9
Ques. Solve log 32 (2 marks)
Ans. Since 32 can be expressed in terms of 25 = 2*2*2*2*2
25 = 32
Hence 5 is the exponent value
So log 32 = 5
Ques. Solve log3(x+1) = 3 (2 Marks)
Ans. We can rewrite the above one as (x+1) = 33
(x+1) = 27
X = 26
Therefore the solution for log3(x+1) = 3 is 26
Ques. log(x+3) + log(x-1) = 1 (2 Marks)
Ans. log[(x+3)(x+1)] = 1
log(x2+4x+4) = 1
(x2+4x+4) = 101
(x2+4x+4) = 0
(x+3)(x+1) = 0
Therefore x =-3, -1
Ques.Solve 6 2x = 4 (2 Marks)
Ans. It can be written in the form of log 6 2x = log 4
2x log 6 = log 4
X = log 4/2 log 6
= 0.6020/2*0.7781
= 1.0823
Therefore the solution is 1.0823
Ques. Prove that log2(x+2)+ log2(4) = log (16) for x =2 (2 Marks)
Ans. Substitute 2 in the above equation we get
LHS = log2(x+2)+ log2(4)
= log2(2+2)+ log2(4)
= log2(4)+ log2(4)
= log2(16)
= RHS
Therefore LHS= RHS
Hence proved
Ques. Find the x value from the given equation log2(2x) = log2(4x+7) (2 Marks)
Ans. Given log2(6x) = log2(4x+2)
6x= 4x+2
2x = 2
X =1
The value of x in the equation log2(2x) = log2(4x+7) is 1.
Ques. Solve log(4x-3)-log(x-4) = log 5 (2 Marks)
Ans. The above equation can be written in the form of
log(4x-3/x-4) = log 5
4x-3/x-4 = 5
4x-3 = 5x-20
X = 17
The x value of the above equation is 17
Ques. Solve 42x+1 = 21 and find the value of x (2 Marks)
Ans. For the above equation apply log on both sides
Log 42x+1 = log 21
(2x+1) log 4 = log 21
2x+1 = (log 21)/(log 4)
= (1.3222)/(0.6020)
2x+1 = 2.19634
2x = 1.1963
X = 0.598
The value of x by solving the above equation is 0.598
Ques. Solve log5(x-10) = 1 (2 Marks)
Ans. The above equation can be written in the form of 5-1 = x-10
5-1 = x-10
5 = x-10
The value of x by solving the above equation is 15
Ques. Express 3logx+8log y = log b in Logarithmic free form. (2 Marks)
Ans. Given 3logx+8log y = log b
Log x3 + log y8 = log b
log(x3 y8) = log b
x3 y8 = b
The above equation in logarithm free form is x3 y8 = b
Ques. Express log10(3)+1 in the form of log10 x (2 Marks)
Ans. The above expression is written in the form of
= log10(3)+1
= log10(3)+ log10(10)
= log10(3*10)
= log10(30)
The above expression in the form of log10 x is log10(30)
Ques. Express \((\frac {1}{3})^4 = \frac {1}{81}\) in logarithmic form. (2 Marks)
Ans. By taking log \(\frac {1}{3}\) of base on both sides, we get
\((\frac {1}{3})^4 = \frac {1}{81}\)
⇒ \(log_{\frac{1}{3}}(\frac {1}{3})^4 =log_{\frac{1}{3}} \frac {1}{81}\)
⇒ \(4log_{\frac{1}{3}}(\frac {1}{3}) =log_{\frac{1}{3}} \frac {1}{81}\) (Since logban = n logba)
⇒ 4 = \(log_{\frac{1}{3}} \frac {1}{81}\) (Since logbb = 1)
⇒ \(log_{\frac{1}{3}} \frac {1}{81}\) = 4
Ques. Find the value of x satisfying log10 (2x + x – 41) = x (1 – log105). (2 Marks)
Ans. We have, log10 (2x + x – 41) = x (1 – log105)
→ log10(2x + x – 41)
→ x log10 2= log10 (2x )
→ 2 x + x – 41 = 2x
→ x = 41.
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