Table of contents  
What are Logarithms?  
Types of Logarithms  
Logarithm Rules  
Logarithm Properties  
Characteristics and Mantissa  
Important Conversions  
Important Formulas  
Solved Examples 
A logarithm is defined as the power to which a number must be raised to get some other values. It is the most convenient way to express large numbers. A logarithm has various important properties that prove multiplication and division of logarithms can also be written in the form of a logarithm of addition and subtraction.
“The logarithm of a positive real number a with respect to base b, a positive real number not equal to 1^{[nb 1]}, is the exponent by which b must be raised to yield a”.
i.e. b^{y}= a ⇔log_{b}a=y
Where,
In other words, the logarithm gives the answer to the question “How many times a number is multiplied to get the other number?”.
For example, how many 3’s are multiplied to get the answer 27?
If we multiply 3 for 3 times, we get the answer 27.
Therefore, the logarithm is 3.
The logarithm form is written as follows:
Log_{3} (27) = 3 ….(1)
Therefore, the base 3 logarithm of 27 is 3.
The above logarithm form can also be written as:
3 x 3 x 3 = 27
3^{3} = 27 …..(2)
Thus, the equations (1) and (2) both represent the same meaning.
Below are some of the examples of conversion from exponential forms to logarithms.
(i) Natural Logarithm: log_{e} N is called natural Logarithm or Naperian Logarithm denoted by (ln N) i.e., when logarithm’s base is “e” then it is called natural logarithm. Ex: log_{e} 7
(ii) Common Logarithm: log_{10} N is called Brigg’s Logarithm when the base is 10. Ex: log_{10 }100
There are certain rules based on which logarithmic operations can be performed. The names of these rules are:
Let us have a look at each of these properties one by one
In this rule, the multiplication of two logarithmic values is equal to the addition of their individual logarithms.
Logb (mn)= logb m + logb n
Example: log3 ( 2y ) = log3 (2) + log3 (y)
The division of two logarithmic values is equal to the difference of each logarithm.
Log_{b} (m/n)= log_{b} m – log_{b} n
Example: log_{3} (2/ y) = log_{3} (2) log_{3} (y)
In the exponential rule, the logarithm of m with a rational exponent is equal to the exponent times its logarithm.
Log_{b} (m^{n}) = n log_{b} m
Example: log_{b}(2^{3}) = 3 log_{b} 2
Log_{b} m = log_{a} m/ log_{a} b
Example: logb 2 = loga 2/loga b
logb (a) = 1 / loga (b)
Example: log_{b} 8 = 1/log_{8} b
If f (x) = log_{b} (x), then the derivative of f(x) is given by;
f'(x) = 1/(x ln(b))
Example: Given, f (x) = log_{10} (x)
Then, f'(x) = 1/(x ln(10))
∫log_{b}(x)dx = x(log_{b}(x) – 1/ln(b)) + C
Example: ∫ log_{10}(x) dx = x ∙ (log_{10}(x) – 1 / ln(10)) + C
Characteristic: The integral part of logarithm is known as characteristic.
Mantissa: The decimal part is known as mantissa and is always positive
In log 3274 = 3.5150, the integral part is 3 i.e., characteristic is 3 and the decimal part is .5150 i.e., mantissa is .5150.
Question 1: If log_{2}X + log_{4}X = log_{0.25}√6 and x > 0, then x is:
A. 6^{1/6}
B. 6^{1/6}
C. 3^{1/3}
D. 6^{1/3}
Correct Answer is Option (A).
Question 2: log_{9} (3log_{2} (1 + log_{3} (1 + 2log_{2}x))) = 1/2. Find x.
A. 4
B. 1/2
C. 1
D. 2
Correct Answer is Option (D).
log_{9} (3log_{2} (1 + log_{3} (1 + 2log_{2}x)) = 1/2
3log_{2}(1 + log_{3}(1 + 2log_{2}x)) = 9^{1/2} = 3
log_{2}(1 + log_{3}(1 + 2log_{2}x) = 1
1 + log_{3}(1 + 2log_{2}x) = 2
log_{3}(1 + 2log_{2}x) = 1
1 + 2log_{2}x = 3
2log_{2}x = 2
log_{2}x = 1
x = 2
The question is "Find x."
Question 3: If 2^{2x+4} – 17 × 2^{x+1} = –4, then which of the following is true?
A. x is a positive value
B. x is a negative value
C. x can be either a positive value or a negative value
D. None of these
Correct Answer is Option (C).
2^{x+4} – 17 * 2^{x+1} = – 4
=> 2^{x+1} = y
2^{2x+2} = y^{2}
2^{2}(2^{2x+2}) – 17 * 2^{x+1} = –4
4y^{2} – 17y + 4 = 0
4y^{2} – 16y – y + y = 0
4y (y – 4) – 1 (y – 4) = 0
y = 1/4 or 4
2^{x+1} = 1/4 or 4
⇒ x + 1 = 2 or – 2
x = 1 or – 3
The question is "which of the following is true?"
Question 4: If log_{12}27 = a, log_{9}16 = b, find log_{8}108
A.
B.
C.
D.
Correct Answer is Option (D).
log_{8}108 = log_{8}(4 * 27)
log_{8}108 = log_{8}4 + log_{8}27
⇒ log_{8}4 = 2/3
log_{8}27 = 2 * log_{16}9
log_{9}16 = b
log_{16}9 = 1/b
log_{8}27 = 2/b
The question is "find log_{8}108."
Question 5: If a, b are integers such that x = a, and x = b satisfy this inequation, find the maximum possible value of a – b.
A. 214
B. 216
C. 200
D. 203
Correct Answer is Option (A).
log_{3}x = y
⇒
y ∈ (3, 5)
3 < log_{3}x < 5
27 < x < 243
Therefore max ( a – b) will be when a = 242 and b = 28. Therefore, max(a – b) = 214.
The question is "find the maximum possible value of a – b."
Question 6: log_{5}x = a (This should be read as log X to the base 5 equals a) log_{20}x = b. What is log_{x}10?
A.
B. (a + b) * 2ab
C.
D.
Correct Answer is Option (A).
Given, log_{5}x = a
log_{20}x = b
log_{x}5 = 1/a
log_{x}20 = 1/b
⇒
The question is "What is log_{x}10?"
Question 7: log_{3}x + log_{x}3 = 17/4. Find x.
A. 34
B. 3^{1/8}
C. 3^{1/4}
D. 3^{1/3}
Correct Answer is Option (C).
log_{3}x + log_{x}3 = 17/4
Let y = log_{3}x
We know that log_{x}3 =
Hence log_{x}3 = 1/y
Thus the equation can be written as
4y^{2} + 4 = 17y
4y^{2} + 4  17y = 0
Solving the above equation we get y = 4 or 1/4
If y = 4
log_{3}x = 4
Then x = 3^{4}
If y = 1/4
log_{3}x = 1/4
Then x = 3^{1/4}
The question is "Find x."
Question 8: log_{x}y + log_{y}x^{2} = 3. Find log_{x}y^{3}.
A. 4
B. 3
C. 3^{1/2}
D. 3^{1/16}
Correct Answer is Option (B).
log_{x}y + log_{y}x^{2} = 3
Let a = log_{x}y
log_{y}x^{2} = 2log_{y}x
We know that log_{y}x =
Hence form above log_{y}x = 1/a
Now rewritting the equation log_{x}y + log_{y}x^{2} = 3
Using a we get
i.e., a^{2}  3a + 2 = 0
Solving we get a = 2 or 1
If a = 2, Then log_{x}y = 2 and log_{y}x^{3} = 3
log_{x}y = 3 * 2 = 6
Or
If a = 1, Then log_{x}y = 1 and log_{y}x^{3} = 3
log_{x}y = 3 * 1 = 3
The question is "Find log_{x}y^{3}."
Question 9: log_{2} 4 * log_{4} 8 * log_{8} 16 * ……………nth term = 49, what is the value of n?
A. 49
B. 48
C. 34
D. 24
Correct Answer is Option (B).
First, the nth term of L.H.S need to be defined by observing the pattern :
It is log_{(2n)} 2.2^{n}
Given,
log_{2} 4 * log_{4} 8 * log_{8} 16 * ……………log_{(2n)} 2.2n = 49
Whenever solving a logarithm equation, generally one should approach towards making the base same.
Making the base 2:
log_{(2n)} 2.2^{n} = 49
log_{(2n)} 2 + log_{(2n)} 2^{n} = 49
1 + n = 49
n = 48
The question is "what is the value of n?"
Question 10: If 3^{3 + 6 + 9 + ……… 3x }= what is the value of x?
A. 3
B. 6
C. 7
D. 11
Correct Answer is Option (D).
First of all, let us define the x^{th} term.
Whenever you encounter a distinctive number such as one given in R.H.S of above equation, always try to find its significance in the context of question.
In this case L.H.S has 3^{a}, so must be some form of 3^{a}.
With little hit and trial, you may find
3^{3(1 + 2 + 3 + ...X)} = 3^{ 3 * 66}
⇒ 3^{3} * 3^{x(x+1)/2} = 3^{3*66}
x(x+1) = 132
Solving this equation for x > 0, we get x = 11.
You should directly be able to see that 132 = 11 * 12 => x= 11
And avoid wasting time solving the complete equation.
The question is "what is the value of x?"
Question 11: x, y, z are 3 integers in a geometric sequence such that y  x is a perfect cube. Given, log_{36}x^{2} + log_{6}√y + log_{216}y^{1/2}z = 6. Find the value of x + y + z.
A. 189
B. 190
C. 199
D. 201
Correct Answer is Option (A).
Let us begin with simplifying the equation:
⇒ log_{62}x^{2} + log_{6}y^{1/2} + 3log_{63}y^{1/2}z = 6
log_{6}x + log_{6}y^{1/2}y^{1/2}z = 6
log_{6}xyz = 6
xyz = 6^{6}
Given x,y,z is in G.P. Let x = a, y = ab, z = ab^{2}
⇒ xyz = a^{3}b^{3} = (ab)^{3}
(ab)^{3} = (6^{2})^{3}
Possible values of (a,b) satisfying the equation :
(1, 36), (2, 18), (3, 12), (4, 9), (9, 4), (12, 3), (18, 2), (36, 1)
Given yx is a perfect cube
⇒ aba is perfect cube
⇒ a(b1) is perfect cube
Only possible when (a, b) = (9, 4)
∴ x = 9 , y = 36 , z = 144
∴ x + y + z = 9 + 36 + 144 = 189
The question is "Find the value of x + y + z."
Question 12: 10^{log(3  10logy)} = log_{2}(9  2^{y}), Solve for y.
A. 0
B. 3
C. 0 and 3
D. none of these
Correct Answer is Option (D).
Before beginning to simplify the equation, don’t forget that anything inside a log cannot be negative
10^{log(3y)} = log_{2}(9  2^{y}) (y > 0)…………………………………(1)
3  y = log_{2}(9  2^{y}) (Therefore, 3  y > 0 =) (y < 3)) ……………………… (2)
2^{3y} = 9  2^{y}
2^{y} = t
⇒ 8 = 9t –t^{2}
⇒ t^{2}  9t + 8 = 0
⇒ t^{2}  t  8t  8 = 0
⇒ t(t  1)  8(t  1) = 0
⇒ t = 1, 8
Therefore, 2^{y} = 1 and 2^{y} = 8
⇒ y = 0 and y = 3
However, from inequalities (1) and (2), y cannot take either of these value.
The question is "Solve for y."
Question 13: 4^{6+12+18+24+…+6x} = (0.0625)^{84}, what is the value of x?
A. 7
B. 6
C. 9
D. 12
Correct Answer is Option (A).
Take right side expression,
= (4^{2})^{84}
= 4^{168}
Take left side expression
4^{6+12+18+24+…+6x} = 4^{6(1+2+3+4+x)}
= 4^{6} * 4^{(1+2+3+4+…+x)}
= 4^{6} * 4^{x(x+1)/2} (using the formula for sum of natural numbers from 1 to x)
Equating left and right side expresssions, we get 4^{6} * 4^{x(x+1)/2} = 4^{168}
Or 4^{6} * 4^{x(x+1)/2} = 4^{6*28}
⇒
or x (x + 1) = 56
Solving for x we get, x = 7
The question is "what is the value of x?"
214 videos139 docs151 tests

1. What are the different types of logarithms? 
2. What are some important logarithm rules to remember? 
3. What are some key logarithm properties to be aware of? 
4. What are characteristics and mantissa in logarithms? 
5. How can logarithms be used for important conversions and calculations? 

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