📌 Big Idea
Because logs and exponents are inverses, the same rules that govern exponents apply to logarithms. These properties let us rewrite log expressions into equivalent forms that are easier to work with.
📐 Properties of Logarithms
Memory trick: Product → logs ADD (like multiplying → add exponents). Quotient → logs SUBTRACT. Power → coefficient moves IN or OUT.
✏️ Example 1 — Condense to a Single Logarithm
a)
log₂ x + log₂ y
↓ Product Property
log₂(xy)
b)
log₃ 5 + log₃ x
↓ Product Property
log₃(5x)
c)
log₁₀ x − log₁₀(5z)
↓ Quotient Property
log(x / 5z)
d)
2 log₅ x − log₅ y
↓ Power then Quotient
log₅(x² / y)
e)
2 log₆ a − 5 log₆ b + log₆ 4
↓ Power + Product + Quotient
log₆(4a² / b⁵)
f)
log(36x)
↓ Already one log — expand via Product
log 36 + log x
✏️ Example 2 — Which expression is equivalent to 2log₃ x − log₃ y?
Step-by-step
2 log₃ x − log₃ y
= log₃(x²) − log₃ y ← Power Property: bring 2 inside
= log₃(x²/y) ← Quotient Property
Answer: log₃(x²/y)
= log₃(x²) − log₃ y ← Power Property: bring 2 inside
= log₃(x²/y) ← Quotient Property
Answer: log₃(x²/y)
Common trap: answer choice log₃(2x/y) is wrong — the 2 is an exponent on x, not a coefficient multiplying x.
✏️ Example 3 — Horizontal Dilation as Vertical Translation
f(x) = log(6x) is a horizontal dilation of g(x) = log x.
Show f(x) = g(x) + k.
f(x) = log(6x)
= log 6 + log x ← Product Property
= log x + log 6
= g(x) + log 6
k = log 6 ≈ 0.778
= log 6 + log x ← Product Property
= log x + log 6
= g(x) + log 6
k = log 6 ≈ 0.778
Key insight: A horizontal dilation by a factor of 1/c is equivalent to a vertical translation by log_b c. This is a powerful connection between transformations!
🔁 Change of Base Property
log_b x = log_a x / log_a b (where a > 0, a ≠ 1)
All logarithmic functions are vertical dilations of each other! You can always convert to base 10 or base e.
✏️ Example 4 — f(x) = log₄ x is a vertical dilation of g(x) = log₉ x
Find k so that f(x) = k · g(x)
f(x) = log₄ x = log₉ x / log₉ 4 ← Change of Base
So k = 1 / log₉ 4
f(x) = (1/log₉ 4) · g(x)
k = 1/log₉ 4
So k = 1 / log₉ 4
f(x) = (1/log₉ 4) · g(x)
k = 1/log₉ 4
Note: log₉ 4 ≈ 0.631, so k ≈ 1.585. The two log functions have the same shape — just stretched/compressed vertically.
🌿 The Natural Logarithm
ln x = loge x (base e ≈ 2.718)
All log properties apply to ln exactly the same way.
✏️ Example 5 — Which expression is equivalent to 3 ln x − 4 ln y?
Step-by-step
3 ln x − 4 ln y
= ln(x³) − ln(y⁴) ← Power Property
= ln(x³/y⁴) ← Quotient Property
Answer: ln(x³/y⁴)
= ln(x³) − ln(y⁴) ← Power Property
= ln(x³/y⁴) ← Quotient Property
Answer: ln(x³/y⁴)
Option A
ln(x³/y⁴)
✓ Correct
Option B
3/4 · ln(x/y)
✗ Wrong
Option C
ln(3x − 4y)
✗ Wrong
Option D
ln(3x/4y)
✗ Wrong
Common mistakes:
• log(x + y) ≠ log x + log y | logs do NOT distribute over addition
• log(x · y) = log x + log y | multiplication → addition
• n · log x = log(xⁿ) | coefficient → exponent (power property)
• log(x/y) = log x − log y | division → subtraction
• log(x + y) ≠ log x + log y | logs do NOT distribute over addition
• log(x · y) = log x + log y | multiplication → addition
• n · log x = log(xⁿ) | coefficient → exponent (power property)
• log(x/y) = log x − log y | division → subtraction
Card 1 of 7
Property
State the Product Property of Logarithms.
Tap to reveal ✨
Answer
log_b(xy) = log_b x + log_b y
Multiplying inside the log → adding two separate logs. Example: log₂(5x) = log₂ 5 + log₂ x.
Topic 2.12
Logarithmic Function Manipulation
0 / 5
Question 1 of 5
Which expression is equivalent to log₄ x + log₄ y?
Question 2 of 5
Which expression is equivalent to 2 log₃ x − log₃ y?
Step 1: Power Property on 2 log₃ x. Step 2: Quotient Property.
Question 3 of 5
Which expression is equivalent to 3 ln x − 4 ln y?
Apply Power Property first, then Quotient Property.
Question 4 of 5
f(x) = log(6x) and g(x) = log x. Express f(x) as a vertical translation of g(x).
Use the Product Property on log(6x).
Question 5 of 5
Which expression is equivalent to 2 log₇ a − 5 log₇ b + log₇ 4?
Apply Power Property to each term, then combine using Product and Quotient.
0/5
Keep going!