4.3Introduction to Logarithmic Functions
Define log_b(x) as the inverse of b^x, convert between exponential and logarithmic form, evaluate common and natural logs, and apply inverse properties.
Logarithms are the mathematical language of scales — pH, decibels, earthquake magnitude, and information theory all use them. They also unlock the ability to solve exponential equations algebraically, making them indispensable in calculus and beyond.
Essential Question
If an exponential function asks "what power gives this result?", how does a logarithm answer that question — and why are the two functions perfect inverses of each other?
Lesson Overview
A logarithm is the inverse of an exponential function. The expression log_b(x) = y asks: "To what power must I raise b to get x?" The answer is y. This single idea unlocks the ability to solve exponential equations, work with pH, decibels, the Richter scale, and much more.
Definition
log_b(x) = y ↔ b^y = x
- b = base (b > 0, b ≠ 1)
- x = argument (must be > 0)
- y = logarithm (the exponent)
Special Logarithms
- Common log: log(x) = log₁₀(x)
- Natural log: ln(x) = log_e(x)
- log_b(1) = 0 (since b⁰ = 1)
- log_b(b) = 1 (since b¹ = b)
- log_b(b^n) = n (inverse property)
- b^(log_b x) = x (inverse property)
Converting between exponential form b^y = x and logarithmic form log_b(x) = y
2ˣ and log₂(x) are inverses — reflected over the line y = x
Worked Examples
Convert each to logarithmic form: (a) 2⁵ = 32 (b) 10³ = 1000 (c) 5⁻² = 1/25
Rule: b^y = x → log_b(x) = y
(a) 2⁵ = 32 → log₂(32) = 5 (base 2, exponent 5, result 32)
(b) 10³ = 1000 → log(1000) = 3 (base 10 = common log)
(c) 5⁻² = 1/25 → log₅(1/25) = −2 (negative exponent is fine)
Convert each to exponential form: (a) log₃(81) = 4 (b) log(0.01) = −2 (c) ln(e³) = 3
Rule: log_b(x) = y → b^y = x
(a) log₃(81) = 4 → 3⁴ = 81 ✓
(b) log(0.01) = −2 → 10⁻² = 0.01 ✓
(c) ln(e³) = 3 → e³ = e³ ✓ (natural log uses base e)
Evaluate without a calculator: (a) log₂(64) (b) log₅(125) (c) log(10,000) (d) log₄(1)
(a) log₂(64): ask "2 to what power = 64?" → 2⁶ = 64 → answer: 6
(b) log₅(125): ask "5 to what power = 125?" → 5³ = 125 → answer: 3
(c) log(10,000): ask "10 to what power = 10,000?" → 10⁴ = 10,000 → answer: 4
(d) log₄(1): ask "4 to what power = 1?" → 4⁰ = 1 → answer: 0
Key fact: log_b(1) = 0 for any valid base b.
Evaluate: (a) log₃(3⁷) (b) 5^(log₅ 12) (c) ln(e) (d) log₂(1/8)
(a) log₃(3⁷) = 7 — inverse property: log_b(b^n) = n
(b) 5^(log₅ 12) = 12 — inverse property: b^(log_b x) = x
(c) ln(e) = log_e(e) = 1 — since e¹ = e
(d) log₂(1/8): 1/8 = 2⁻³, so log₂(2⁻³) = −3
Solve for x: (a) log₄(x) = 3 (b) log_x(36) = 2 (c) log₂(x) = −4
(a) log₄(x) = 3 → 4³ = x → x = 64
(b) log_x(36) = 2 → x² = 36 → x = 6 (base must be positive, not 1)
(c) log₂(x) = −4 → 2⁻⁴ = x → x = 1/16
Strategy: convert to exponential form, then solve for the unknown.
Guided Practice
Convert to logarithmic form: (a) 4³ = 64 (b) e² ≈ 7.389 (c) 7⁻¹ = 1/7
Hint: Use the rule b^y = x → log_b(x) = y. Identify base, exponent, and result.
Convert to exponential form: (a) log₆(216) = 3 (b) ln(1) = 0 (c) log(100) = 2
Hint: Use log_b(x) = y → b^y = x. For ln, the base is e; for log, the base is 10.
Evaluate without a calculator: (a) log₃(27) (b) log₁₀(0.001) (c) log₅(5) (d) log₂(1/4)
Hint: Ask 'base to what power equals the argument?' Use b⁰=1, b¹=b, and negative exponents for fractions.
Use inverse properties to evaluate: (a) log₇(7⁵) (b) 4^(log₄ 9) (c) ln(e⁻³)
Hint: Inverse properties: log_b(b^n) = n and b^(log_b x) = x. Apply directly.
Solve for x: (a) log₃(x) = 4 (b) log₂(x) = −3 (c) log_x(49) = 2
Hint: Convert to exponential form first: log_b(x) = y → b^y = x.
Key Vocabulary
Logarithm
The exponent to which a base must be raised to produce a given number. log_b(x) = y means b^y = x.
Base of a Logarithm
The number b in log_b(x). Must satisfy b > 0 and b ≠ 1. Determines the 'scale' of the logarithm.
Common Logarithm
A logarithm with base 10, written log(x) or log₁₀(x). Used in pH, decibels, and the Richter scale.
Natural Logarithm
A logarithm with base e ≈ 2.718, written ln(x). Fundamental in calculus and continuous growth models.
Argument
The input x in log_b(x). The argument must always be strictly positive (x > 0).
Inverse Functions
Two functions that undo each other. f(x) = b^x and g(x) = log_b(x) are inverses: f(g(x)) = x and g(f(x)) = x.
Inverse Properties
log_b(b^n) = n and b^(log_b x) = x. These follow directly from the definition of inverse functions.
Vertical Asymptote
For f(x) = log_b(x), the y-axis (x = 0) is a vertical asymptote — the graph approaches but never crosses it.
Check Your Understanding
Interactive Practice — 5 Questions
Which exponential equation is equivalent to log₅(125) = 3?
Evaluate log₂(32).
What is the value of log₉(1)?
Evaluate 6^(log₆ 17).
Solve: log₃(x) = −2.
Independent Practice
Independent Practice
Convert to logarithmic form: (a) 6² = 36 (b) 10⁻³ = 0.001 (c) e⁰ = 1
Convert to exponential form: (a) log₄(64) = 3 (b) log(10) = 1 (c) ln(e⁵) = 5
Evaluate without a calculator: (a) log₂(128) (b) log₃(1/9) (c) log(1,000,000) (d) ln(e)
Use inverse properties: (a) log₅(5⁸) (b) 3^(log₃ 20) (c) log₁₀(10^(−4))
Solve for x: (a) log₅(x) = 3 (b) log_x(64) = 3 (c) log₂(x) = 0
Common log: log(x) = log₁₀(x) — passes through (1,0) and (10,1)
Natural log: ln(x) — passes through (1,0) and (e,1)
Common Mistakes
Writing log₂(8) = 2 because "log₂ means divide by 2".
log₂(8) = 3 because 2³ = 8. A logarithm is an exponent, not a division.
Evaluating log(−5) or log(0) — treating negative/zero arguments as valid.
The argument of a logarithm must be strictly positive (x > 0). log(−5) and log(0) are undefined.
Confusing log_b(b) = b with the correct value.
log_b(b) = 1 always, because b¹ = b. Similarly, log_b(1) = 0 because b⁰ = 1.
Forgetting to convert to exponential form when solving log_b(x) = y for x.
Always convert first: log_b(x) = y → b^y = x. Then solve for the unknown.
Math Tips
"A logarithm is an exponent." Repeat this until it's automatic — it's the entire definition.
The two inverse properties are your fastest tools: log_b(b^n) = n and b^(log_b x) = x. Memorize both.
To evaluate log_b(x) mentally: ask "b to what power gives x?" Write it as b^? = x and solve.
log(x) always means base 10 on calculators and in most textbooks. ln(x) always means base e.
Negative logarithms are valid — they just mean the argument is between 0 and 1. log₂(1/8) = −3 because 2⁻³ = 1/8.