Equations and inequalities

Exponential and log equations

When x sits in the exponent, you either write both sides with the same base or take the logarithm. Log equations work the other way round.

The rules

• Same base: aˣ = aʸ means x = y. Write 8 as 2³, 27 as 3³ and so on.
• Otherwise take the logarithm: aˣ = b gives x = log(b) / log(a).
• Sums of powers: factor out the smallest power: 2ˣ⁺¹ + 2ˣ = 2ˣ(2 + 1).
• Log equations: logₐ(x) = c gives x = aᶜ; the argument must be positive.

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. Solve for x: 5ˣ = 25

   • a2
   • b1
   • c5
   • d3
   Show solution

   Answer: 2

   Write the right side as a power of the same base, then match exponents.

   25 = 5², so x = 2.

2. 2. Solve for x: 3ˣ = 243

   • a4
   • b6
   • c3
   • d5
   Show solution

   Answer: 5

   Write the right side as a power of the same base, then match exponents.

   243 = 3⁵, so x = 5.

3. 3. Solve for x: log₁₀(x) = 5

   • a100010
   • b100000
   • c50
   • d99990
   Show solution

   Answer: 100000

   logₐ(x) = e means x = aᵉ.

   x = 10⁵ = 100000.

4. 4. Solve for x: log₂(x) = 4

   • a16
   • b14
   • c8
   • d18
   Show solution

   Answer: 16

   logₐ(x) = e means x = aᵉ.

   x = 2⁴ = 16.

5. 5. Solve for x: log₁₀(x) = 3

   • a990
   • b1000
   • c30
   • d1010
   Show solution

   Answer: 1000

   logₐ(x) = e means x = aᵉ.

   x = 10³ = 1000.

6. 6. Solve for x: log₃(x) = 5

   • a246
   • b240
   • c15
   • d243
   Show solution

   Answer: 243

   logₐ(x) = e means x = aᵉ.

   x = 3⁵ = 243.

Common mistakes

• Taking the log of each term of a sum separately.
• Forgetting that aˣ is always positive, so aˣ = −4 has no solution.
• Not checking that log arguments stay positive.

Related topics

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