Every operation has an inverse. We have subtraction for addition, division for multiplication, cube roots for cubing โ and logarithms for exponentiation. Just like 0.5 = ยฝ, an equation can be written in exponential or logarithmic form โ both say the same thing.
โ๏ธ Inverse Operations
Operation
Example
Inverse Operation
Addition
x + 3 = 7
Subtraction
Multiplication
3x = 7
Division
Cubing
xยณ = 7
Cube Root
Exponentiation
3หฃ = 7
Logarithm โ new!
๐ Converting Between Exponential and Log Form
bc = a โบ logb a = c
These two forms say the SAME thing. b = base, c = exponent/power, a = result.
Special Notation:
โข Common log: log(x) means log10(x) โ base 10 is understood
โข Natural log: ln(x) means loge(x) โ base e โ 2.718
โ๏ธ Example 1 โ Exponential โ Log Form
Exponential Form
Log Form
a)
3ยฒ = 9
โ
log3 9 = 2
b)
10x = y
โ
log y = x (base 10)
c)
ex = y
โ
ln y = x (base e)
d)
y = e3x
โ
ln y = 3x
โ๏ธ Example 2 โ Log Form โ Exponential Form
Log Form
Exponential Form
a)
log7 y = 2
โ
7ยฒ = y
b)
y = log3 x
โ
3y = x
c)
log y = 2
โ
10ยฒ = y (base 10)
d)
ln y = 4
โ
e4 = y (base e)
๐งฎ Evaluating Logs Without a Calculator
Strategy: Ask "what power of b gives me a?" Convert to exponential form to figure it out.
โ๏ธ Example 3 โ Evaluate Without a Calculator
log2 8
2? = 8 โ 2ยณ = 8
= 3
log5 25
5? = 25 โ 5ยฒ = 25
= 2
log2 32
2? = 32 โ 2โต = 32
= 5
log 100
10? = 100 โ 10ยฒ = 100
= 2
log16 4
16? = 4 โ 16ยฝ = 4
= ยฝ
log3 1
3? = 1 โ 3โฐ = 1
= 0
log 10
10? = 10 โ 10ยน = 10
= 1
log6 (1/36)
6? = 1/36 โ 6โปยฒ = 1/36
= โ2
Useful patterns to memorize:
logb 1 = 0 always (bโฐ = 1) | logb b = 1 always (bยน = b) | Negative log result โ b was raised to a negative power
๐ข Example 4 โ Using a Calculator
Evaluate with โฅ 4 decimal places
log 9
log base 10
โ 0.9542
log 6125
log base 10
โ 3.7871
log2 10
change of base: log10/log2
โ 3.3219
log7 135
change of base: log135/log7
โ 2.5208
Change of Base Formula: logb a = log(a) / log(b) = ln(a) / ln(b)
๐ Logarithmic Scales
Why Use a Log Scale?
Exponential data can span many orders of magnitude. On a normal scale, small values become indistinguishable when large values dominate. A log scale spaces gridlines at powers of 10, making all data points visible and comparable.
Normal scale: 1, 2, 3, 4, 5, 6, 7โฆ (equal spacing by +1)
Log scale: 10ยน, 10ยฒ, 10ยณ, 10โด, 10โตโฆ (equal spacing by ร10)
โ๏ธ Example 5 โ Plotting on a Log Scale
Population Data (1840โ1880)
Year
1840
1860
1880
Population
17,069,453
31,443,321
50,189,209
log(Population)
7.2322โฆ
7.4975โฆ
7.7006โฆ
To plot on a log scale: take log(value) to find the position. 17,069,453 โ 107.23, so it plots between the 10โท and 10โธ gridlines.
On a log scale, the y-value 2 (point C) is near the bottom while 350 (point E) is near the top โ all values are visible despite spanning nearly 3 orders of magnitude.
Card 1 of 7
Concept
What is the fundamental relationship between b^c = a and log form?
Tap to reveal โจ
Answer
b^c = a โบ log_b(a) = c
Both forms say the same thing. b = base, c = exponent/power, a = result. Switching between them is just rewriting, not changing the equation.
Topic 2.9
Logarithmic Expressions
0 / 5
Question 1 of 5
Convert 3ยฒ = 9 to logarithmic form.
Question 2 of 5
Convert log7 y = 2 to exponential form.
Question 3 of 5
Evaluate log2 32 without a calculator.
Ask: 2 to what power equals 32?
Question 4 of 5
Evaluate log6(1/36) without a calculator.
Ask: 6 to what power equals 1/36?
Question 5 of 5
Why is a logarithmic scale useful for displaying exponential data?
0/5
Keep going!
๐ฌ Ask a Question
Stuck on something? Your teacher will reply to your email!