A logarithm answers the question: "what power do I need to raise this base to, to get this number?" If bˣ = y, then logᵦ(y) = x — logarithms and exponents are inverse operations. For example, since 2³ = 8, log₂(8) = 3. Logarithms are especially useful for solving equations where the unknown is an exponent.
Example
If a population growth equation is 100(2)ᵗ = 3200, solving for t requires logarithms: dividing both sides gives 2ᵗ = 32, and since 2⁵ = 32, t = 5 — so log₂(32) = 5.
Key terms
Logarithm:
The power needed to raise a base to, to reach a given number.
Base:
The number being raised to a power in a logarithm or exponent.
Inverse operation:
An operation that reverses another; logarithms reverse exponents.
Questions
1. A logarithm answers the question:
What power do I raise the base to, to get this number?
What number do I add to reach zero?
What is the square root of a number?
What is the sum of two numbers?
2. Since 2³ = 8, log₂(8) equals:
3
8
2
6
3. Logarithms and exponents are:
Inverse operations
Completely unrelated operations
The exact same operation with no difference
Only used in geometry
4. Since 10² = 100, log₁₀(100) equals:
2
100
10
20
5. In logᵦ(y) = x, "b" represents:
The base
The final answer only
A type of punctuation
The exponent only
6. Since 3² = 9, log₃(9) equals:
2
9
3
6
7. Logarithms are especially useful for solving equations where:
The unknown is an exponent
There is no unknown at all
Only addition is involved
The equation has no solution ever
8. Since 2⁵ = 32, log₂(32) equals:
5
32
2
10
9. If 100(2)ᵗ = 3200, then 2ᵗ equals:
32
16
64
3200
10. Since 5³ = 125, log₅(125) equals:
3
125
5
15
11. Since 4² = 16, log₄(16) equals:
2
16
4
8
12. For the equation 100(2)ᵗ = 3200, solving for t gives:
5
3
16
32
13. Why are logarithms described as the "inverse" of exponents, similar to how division is the inverse of multiplication?
Each undoes what the other does — a logarithm can recover the exponent an exponential expression produced
Logarithms and exponents always produce the exact same result
Inverse operations never appear anywhere else in mathematics
Logarithms have no mathematical relationship to exponents at all
14. Since 10³ = 1000, log₁₀(1000) equals:
3
1000
10
30
15. Why can't you solve an equation like 2ˣ = 50 using only basic algebra (without logarithms)?
Basic algebra isolates variables through direct operations, but x is in the exponent position, requiring a logarithm to isolate it
Equations with exponents can always be solved with simple subtraction
This type of equation has no possible solution
Basic algebra and logarithms solve exponent equations identically
16. If a bacteria population models growth as P(t) = 50(3)ᵗ and reaches 1350, approximately what is t?
3
2
4
27
17. Why are logarithms often used in real-world contexts like measuring earthquake magnitude (the Richter scale) or sound (decibels)?
These phenomena vary across a huge range of values, and logarithms compress that range into a more manageable scale
Logarithms have no practical real-world applications
Earthquake and sound measurements never involve any mathematical scaling
The Richter and decibel scales use only simple addition, not logarithms
18. If log₂(x) = 6, what is x?
64
12
32
36
19. Why might understanding logarithms help when working with compound interest or investment growth problems?
They allow you to solve for time when the interest rate and target amount are known but time is the unknown exponent
Logarithms have no application in financial mathematics
Compound interest problems never involve exponents
Logarithms can only be used with population growth, not finance
20. If log₅(x) = 3, what is x?
125
15
25
8
21. Why is log₂(0) undefined, unlike log₂(1) which equals 0?
No power of 2 can ever produce 0, since any power of a positive base is always positive
Log₂(0) is defined and simply equals negative infinity with no issue
Zero can always be reached by raising 2 to some specific power
Undefined logarithms only occur with bases other than 2
Answer key (parent copy)
1. What power do I raise the base to, to get this number?
2. 3
3. Inverse operations
4. 2
5. The base
6. 2
7. The unknown is an exponent
8. 5
9. 32
10. 3
11. 2
12. 5
13. Each undoes what the other does — a logarithm can recover the exponent an exponential expression produced
14. 3
15. Basic algebra isolates variables through direct operations, but x is in the exponent position, requiring a logarithm to isolate it
16. 3
17. These phenomena vary across a huge range of values, and logarithms compress that range into a more manageable scale
18. 64
19. They allow you to solve for time when the interest rate and target amount are known but time is the unknown exponent
20. 125
21. No power of 2 can ever produce 0, since any power of a positive base is always positive