Mathematics ยท Quantitative Aptitude

Logarithms

246 Questions

Logarithms are mathematical operations that determine the exponent required for a base to reach a specific number. This topic tests the application of logarithmic properties, changing bases, and solving complex equations. It is a high-yield topic for quantitative aptitude in competitive exams.

Logarithmic expressionsBase change propertiesSolving log equationsInfinite series logsCharacteristic values

Logarithms Questions

Multiple choice

If log(x + 1) - log(x - 1) = 2, what is the value of x?

  1. 2

  2. 3

  3. 4

  4. 5
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

log(x + 1) - log(x - 1) = 2 log((x + 1)/(x - 1)) = 2 (x + 1)/(x - 1) = 10^2 x + 1 = 100(x - 1) x + 1 = 100x - 100 99x = 101 x = 101/99 = 3.

Multiple choice

If log(x + 1) - log(x - 1) = 2, what is the value of x?

  1. 2

  2. 3

  3. 4

  4. 5
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

log(x + 1) - log(x - 1) = 2 log((x + 1)/(x - 1)) = 2 (x + 1)/(x - 1) = 10^2 x + 1 = 100(x - 1) x + 1 = 100x - 100 99x = 101 x = 101/99 = 3.

Multiple choice

What is the value of (\log_2 32)?

  1. 4

  2. 5

  3. 6

  4. 7
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

To evaluate (\log_2 32), we need to find the exponent to which 2 must be raised to get 32. We can write (32 = 2^x). Solving for x, we get: (x = \log_2 32). Therefore, the value of (\log_2 32) is 5.

Multiple choice

What is the value of (log_{10} 100)?

  1. 1

  2. 2

  3. 10

  4. 100
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using the logarithmic property (log_{b} b^n = n), we have (log_{10} 100 = log_{10} (10^2) = 2).

Multiple choice

Solve the equation (log_2 (x + 3) = 5).

  1. \(x = 27\)
  2. \(x = 31\)
  3. \(x = 33\)
  4. \(x = 35\)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Rewrite the equation as (2^5 = x + 3), then solve for (x) to get (x = 31).

Multiple choice

Find the value of (log_5 (1/125)).

  1. \(-3\)
  2. \(-2\)
  3. \(-1\)
  4. \(0\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the logarithmic property (log_b (1/a) = - log_b a), we have (log_5 (1/125) = log_5 (5^{-3}) = -3).

Multiple choice

Which of the following is equivalent to (log_a (b/c))?

  1. \(log_a b - log_a c\)
  2. \(log_a b + log_a c\)
  3. \(log_a b / log_a c\)
  4. \(log_a c - log_a b\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the logarithmic property (log_b (a/c) = log_b a - log_b c), we have (log_a (b/c) = log_a b - log_a c).

Multiple choice

Solve the equation (log_3 (2x - 1) = 2).

  1. \(x = 3\)
  2. \(x = 4\)
  3. \(x = 5\)
  4. \(x = 6\)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Rewrite the equation as (3^2 = 2x - 1), then solve for (x) to get (x = 5).

Multiple choice

Find the value of (log_2 (1/16)).

  1. \(-4\)
  2. \(-3\)
  3. \(-2\)
  4. \(-1\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the logarithmic property (log_b (1/a) = - log_b a), we have (log_2 (1/16) = log_2 (2^{-4}) = -4).

Multiple choice

Which of the following is equivalent to (log_a (b^2 / c^3))?

  1. \(2 log_a b - 3 log_a c\)
  2. \(2 log_a b + 3 log_a c\)
  3. \(log_a b^2 - log_a c^3\)
  4. \(log_a b^2 / log_a c^3\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the logarithmic property (log_b (a/c) = log_b a - log_b c), we have (log_a (b^2 / c^3) = log_a b^2 - log_a c^3 = 2 log_a b - 3 log_a c).

Multiple choice

Find the value of (log_3 (1/27)).

  1. \(-3\)
  2. \(-2\)
  3. \(-1\)
  4. \(0\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the logarithmic property (log_b (1/a) = - log_b a), we have (log_3 (1/27) = log_3 (3^{-3}) = -3).

Multiple choice

Which of the following is equivalent to (log_a (b^3 c^2 / d^4))?

  1. \(3 log_a b + 2 log_a c - 4 log_a d\)
  2. \(3 log_a b + 2 log_a c + 4 log_a d\)
  3. \(log_a b^3 + log_a c^2 - log_a d^4\)
  4. \(log_a b^3 c^2 / log_a d^4\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the logarithmic property (log_b (a/c) = log_b a - log_b c), we have (log_a (b^3 c^2 / d^4) = log_a b^3 + log_a c^2 - log_a d^4 = 3 log_a b + 2 log_a c - 4 log_a d).

Multiple choice

Solve the equation (log_5 (4x - 3) = 2).

  1. \(x = 7\)
  2. \(x = 8\)
  3. \(x = 9\)
  4. \(x = 10\)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Rewrite the equation as (5^2 = 4x - 3), then solve for (x) to get (x = 9).

Multiple choice

What does the symbol (\log) represent?

  1. Logarithm

  2. Exponential

  3. Sine

  4. Cosine
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

(\log) is a mathematical symbol that represents the logarithm of a number. It is used to find the exponent to which a base number must be raised to produce a given number.

Multiple choice

What is the value of x in the equation log(x + 1) = 2?

  1. 9

  2. 10

  3. 11

  4. 12
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To solve the equation log(x + 1) = 2, we need to rewrite the equation in exponential form. The exponential form of the equation is x + 1 = 10^2 = 100. Subtracting 1 from both sides of the equation, we get x = 99. Therefore, the value of x is 9.