A logarithm represents the exponent to which a number must be raised to yield another number. The majority of aptitude questions related to logarithms rely on formulas.
log_{b} (x) if and only if b^{y} = x
Common logarithm has base 10 (b = 10) and is denoted as log(x), while natural logarithm has base e (Euler’s number) and is denoted as ln(x).
A logarithm is expressed as the inverse of exponentiation. In simpler terms, when we take the logarithm of a specific value, we essentially reverse the process of exponentiation. Here are discussions and answers related to logarithm questions:
For example: If we choose a base such as b = 3 and raise it to the power of k = 2, the result is 32, denoted by C, expressed as 32 = C. The rules of exponentiation can be applied to deduce that the result is C = 32 = 8.
As an illustration, consider someone asking, "To which power must 2 be raised to equal 16?" The answer is 4. This can be expressed through logarithmic calculation, i.e., log2(16) = 4, which is articulated as "log base two of sixteen is four."
Generalizing the examples above leads us to the formal definition of a logarithm.
Logb (a) =c ↔ bc =a
Both the equations define the similar link where:
‘b’ is considered as the base,
c is considered as the exponent
a is considered as the argument
Example 1: If the value of log 3 = 0.477, then find the number of digits in 3^{36}
(a) 18
(b) 3
(c) 20
(d) 24
Ans: (b)
Number of digits in 336 ≈ ⌊log10(336)⌋ + 1
Number of digits in 336 ≈ ⌊log(112 * 3)⌋ + 1
Number of digits in 336 ≈ ⌊(log(112) + log(3))⌋ + 1
Number of digits in 336 ≈ ⌊(2.049 + 0.477)⌋ + 1
Number of digits in 336 ≈ ⌊2.526⌋ + 1
Number of digits in 336 ≈ 2 + 1
Number of digits in 336 ≈ 3
So, the number 336 has 3 digits.
Example 2: The value of log_{3} 27 is :
(a) 3
(b) 2
(c) 7
(d) 8
Ans: (a)
log_{3} 3^{3}
= 3
Example 3: Solve : p^{x} = q^{y}
(a) z/x
(b) y/x
(c) X/z
(d)A/x
Ans: (b)
log p^{x} = log q^{y}
x log p = y log q
log p / log q = y/x
Example 4: Solve: log √9 /log 9
(a) zero
(b) 1/2
(c) 1
(d) 1/3
Ans: (b)
(log 9^{1/2})/ log 9
= 1/2(log 9/ log 9)
= 1/2
Example 5: Solve: log_{6}x^{3} = 18
(a) 46656
(b) 4/9
(c) 223456
(d) 4/6
Ans: (a)
x^{3} = 6^{18}
x = 6^{6}
x = 46656
Example 6: If log_{x} (5/18) = 1/2 , then find the value of x:
(a) 456/12
(b) 25/324
(c) 324/78
(d) 566/18
Ans: (b)
log_{x}5/18 = 1/2
x^{1/2} = 5/18
√x = 5/18
Squaring both sides
x = (5/18)^{2}
x = 25/324
Example 7: If log 5 = 0.698, find the number of digits in 5^{25}
(a) 3
(b) 18
(c) 4
(d) 6
Ans: (b)
log (5^{25}) = 25* log 5
= 25* 0.698
= 17.45
= 18 (approx.)
Example 8: Solve the given logarithmic equation: log_{7}x = 3
(a) 324
(b) 343
(c) 289
(d) 366
Ans: (b)
Taking base 7 antilog on both sides,
log_{7}x = 3
⇒ X = 7^{3}
⇒ X = 343
Example 9: Solve : log_{x}√3 = 1/2
(a) 3
(b) 2
(c) 4
(d) 6
Ans: (a)
On evaluating the given equation,
x > 0 , x 1
x^{1/2} = √3
x^{1/2} = √3
squaring both side
(x^{1/2})^{2} = √3^{2}
x = 3
Example 10: Prove : log_{6}36 = 3x
(a) 2/3
(b) 4/5
(c) 6/7
(d) 6/8
Ans: (a)
log_{6}36 = log_{6}(6)^{2}
2log_{6}6 = 2 ( logaa = 1)
2 = 3x
x = 2/3
314 videos170 docs185 tests

1. What is a logarithm and how is it related to exponents? 
2. How do logarithms simplify calculations? 
3. What are the properties of logarithms? 
4. How can logarithms be used in reallife applications? 
5. Can logarithms be negative? 
314 videos170 docs185 tests


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