1/12
2.9 · Core Relationship
State the relationship between bc = a and logarithmic form.
2/12
2.9 · Special Notation
What does log x mean (no base shown)? What does ln x mean?
What base is "understood" in each case?
3/12
2.9 · Convert: Exp → Log
Convert 32 = 9 to logarithmic form.
bc = a → logb a = c
4/12
2.9 · Convert: Exp → Log
Convert 10x = y to logarithmic form.
5/12
2.9 · Convert: Log → Exp
Convert log7 y = 2 to exponential form.
logb a = c → bc = a
6/12
2.9 · Convert: Log → Exp
Convert ln y = 4 to exponential form.
7/12
2.9 · Evaluate: No Calc
Evaluate log2 32 without a calculator.
2 to what power equals 32?
8/12
2.9 · Evaluate: No Calc
Evaluate log5 25 without a calculator.
5 to what power equals 25?
2/12
✓ Special Notation
log x = log10 x (base 10) ln x = loge x (base e ≈ 2.718)
The common log omits the base by convention. The natural log uses "ln" notation.
1/12
✓ Core Relationship
bc = a ⇔ logb a = c
b = base, c = exponent, a = result. Both forms say the same thing — just rewritten differently.
4/12
✓ Convert: 10x = y
log y = x
Base 10 → common log (no base written). b=10, c=x, a=y. So log10 y = x → log y = x.
3/12
✓ Convert: 32 = 9
log3 9 = 2
b=3, c=2, a=9. Plug into logb a = c: log3 9 = 2.
6/12
✓ Convert: ln y = 4
e4 = y
ln = loge. So b=e, c=4, a=y. Exponential form: e4 = y.
5/12
✓ Convert: log7 y = 2
72 = y
b=7, c=2, a=y. Exponential form: 72 = y = 49.
8/12
✓ Evaluate: log5 25
2
52 = 25. So log5 25 = 2.
7/12
✓ Evaluate: log2 32
5
25 = 32. So log2 32 = 5. Strategy: ask "what power of 2 gives 32?"
9/12
2.9 · Evaluate: No Calc
Evaluate log3 1 without a calculator. Explain why.
10/12
2.9 · Evaluate: Negative Result
Evaluate log6(1/36) without a calculator.
6 to what power gives 1/36?
11/12
2.9 · Patterns
What are the values of logb 1 and logb b for any base b?
12/12
2.9 · Log Scale
Why is a logarithmic scale useful for displaying exponential data?
10/12
✓ Negative Result
−2
6−2 = 1/62 = 1/36. So log6(1/36) = −2. A negative result means the argument was less than 1.
9/12
✓ Evaluate: log3 1
0 — because 30 = 1
Any base raised to the 0 power = 1. So logb 1 = 0 for any valid base b.
12/12
✓ Log Scale
Allows very large and very small values to all be visible and comparable
Log scales space gridlines at powers of 10. Exponential data spread over many orders of magnitude becomes readable.
11/12
✓ Key Patterns
logb 1 = 0 logb b = 1
b0 = 1 always → logb 1 = 0. b1 = b always → logb b = 1. Memorize these!