Arithmetic and terms

Logarithms

A logarithm answers one question: to which power must I raise the base to get this number? log₂(8) = 3 because 2³ = 8. With that and three rules you can solve every log task in an entrance test.

The rules

• Definition: logₐ(x) = y means aʸ = x.
• Product: logₐ(xy) = logₐ(x) + logₐ(y); quotient: logₐ(x/y) = logₐ(x) − logₐ(y).
• Power: logₐ(xⁿ) = n · logₐ(x).
• logₐ(1) = 0 and logₐ(a) = 1; change of base: logₐ(x) = log(x) / log(a).

Practice questions with solutions

Generated by the same engine as the Kollego course and solvable without a calculator. Pick an answer in your head, then open the solution.

1. 1. Compute without a calculator: log₅(√5)

   • a1.0
   • b0.5
   • c1.5
   • d2.0
   Show solution

   Answer: 0.5

   A root is a fractional power: ᵏ√(aⁿ) is a to the power n/k, and logₐ(aˣ) = x.

   √5 = 5^(1/2) ⇒ log_5 = 0.5.

2. 2. Compute without a calculator: log₂(√32)

   • a0.4
   • b10.0
   • c2.5
   • d5.0
   Show solution

   Answer: 2.5

   A root is a fractional power: ᵏ√(aⁿ) is a to the power n/k, and logₐ(aˣ) = x.

   √32 = 2^(5/2) ⇒ log_2 = 2.5.

3. 3. Evaluate: log₂(4)

   • a3
   • b2
   • c4
   • d1
   Show solution

   Answer: 2

   logₐ(x) asks: to what power must a be raised to get x?

   2² = 4, so log₂(4) = 2.

4. 4. Evaluate: log₁₀(100000)

   • a6
   • b4
   • c10
   • d5
   Show solution

   Answer: 5

   logₐ(x) asks: to what power must a be raised to get x?

   10⁵ = 100000, so log₁₀(100000) = 5.

5. 5. Evaluate: log₅(625)

   • a3
   • b4
   • c125
   • d5
   Show solution

   Answer: 4

   logₐ(x) asks: to what power must a be raised to get x?

   5⁴ = 625, so log₅(625) = 4.

6. 6. Evaluate: log₂(32)

   • a6
   • b4
   • c5
   • d2
   Show solution

   Answer: 5

   logₐ(x) asks: to what power must a be raised to get x?

   2⁵ = 32, so log₂(32) = 5.

Common mistakes

• Splitting a sum: log(x + y) is not log(x) + log(y).
• Forgetting that the argument must be positive.
• Mixing up base and argument when reading logₐ(x).

Related topics

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