Logarithms & Exponential Equations
Definition, all 6 log laws, solving exponential and logarithmic equations, graphs of log and exponential functions, and real-world applications. Everything you need for precalculus Chapter 4.
Quick Reference — All 6 Log Laws
Product Rule
log(ab) = log a + log b
Multiplication inside → addition outside
Quotient Rule
log(a/b) = log a − log b
Division inside → subtraction outside
Power Rule
log(aⁿ) = n · log a
Exponent inside → multiplier outside
Change of Base
log_b(x) = ln(x) / ln(b)
Convert any base using ln or log
Log of Its Own Base
log_b(b) = 1
Because b¹ = b
Log of 1
log_b(1) = 0
Because b⁰ = 1 for any base b
Key Graph Facts — y = bˣ
- Domain: all reals · Range: (0, ∞)
- y-intercept: (0, 1) · Asymptote: y = 0
- Increasing if b > 1, decreasing if 0 < b < 1
Key Graph Facts — y = log_b(x)
- Domain: (0, ∞) · Range: all reals
- x-intercept: (1, 0) · Asymptote: x = 0
- Inverse of y = bˣ — reflection over y = x
What Is a Logarithm?
A logarithm is an exponent — that is the single most important thing to remember. The logarithmic form and the exponential form say exactly the same thing, just written differently:
logb(x) = y ⟺ by = x
The logarithm y is the exponent to which you raise base b to get x.
The three most common bases in precalculus are:
Base 10
log(x)
Common logarithm — the default when no base is written. Used in pH, decibels, Richter scale.
Base e
ln(x)
Natural logarithm — base is Euler's number e ≈ 2.718. Essential for calculus and growth/decay problems.
Base 2
log₂(x)
Binary logarithm — used in computer science and information theory.
Practice converting between forms
log₂(8) = 3 ⟺ 2³ = 8
log(1000) = 3 ⟺ 10³ = 1000
ln(e²) = 2 ⟺ e² = e²
log₅(1) = 0 ⟺ 5⁰ = 1
The 6 Logarithm Laws
These laws hold for any valid base b (> 0, b ≠ 1) and any positive arguments. They let you expand, condense, or rewrite logarithmic expressions to solve equations.
| Law | Formula | Example | Memory cue |
|---|---|---|---|
| Product Rule | log(ab) = log a + log b | log(100·10) = log 100 + log 10 = 2 + 1 = 3 | Multiplication inside → addition outside |
| Quotient Rule | log(a/b) = log a − log b | log(1000/10) = log 1000 − log 10 = 3 − 1 = 2 | Division inside → subtraction outside |
| Power Rule | log(aⁿ) = n · log a | log(10⁵) = 5 · log 10 = 5 · 1 = 5 | Exponent inside → multiplier outside |
| Change of Base | log_b(x) = ln(x) / ln(b) | log_2(10) = ln(10)/ln(2) ≈ 3.322 | Convert any base using ln or log |
| Log of Its Own Base | log_b(b) = 1 | log_3(3) = 1, ln(e) = 1, log(10) = 1 | Because b¹ = b |
| Log of 1 | log_b(1) = 0 | log(1) = 0, ln(1) = 0, log_7(1) = 0 | Because b⁰ = 1 for any base b |
Domain warning
Log laws only apply when all arguments are positive. You cannot take the log of a negative number or zero — these are outside the domain of any logarithmic function. Always check domain restrictions before applying log laws.
Solving Exponential Equations — 4 Methods
An exponential equation has the variable in the exponent, such as 3ˣ = 20. The method you use depends on whether the bases can be made equal.
Same Base — Equate Exponents
When both sides can be written with the same base, just set the exponents equal. This works when the numbers are recognizable powers (like 4, 8, 27, 125, etc.).
Solve: 2ˣ = 2⁵
Same base on both sides → x = 5
Solve: 4ˣ = 8
Rewrite: (2²)ˣ = 2³ → 2²ˣ = 2³
Equate exponents: 2x = 3 → x = 3/2
Take Log of Both Sides
When bases cannot be matched, take the natural log (or log base 10) of both sides, then use the power rule to bring the exponent down. This is the most general method.
Solve: 3ˣ = 20
Take ln of both sides: ln(3ˣ) = ln(20)
Power rule: x · ln(3) = ln(20)
Divide: x = ln(20) / ln(3) ≈ 2.727
Substitution — Quadratic Form
When the equation is quadratic in an exponential expression, substitute u = bˣ to convert it into a regular quadratic, solve for u, then recover x.
Solve: 4ˣ − 5 · 2ˣ + 6 = 0
Note: 4ˣ = (2²)ˣ = (2ˣ)². Let u = 2ˣ
Substitute: u² − 5u + 6 = 0
Factor: (u − 2)(u − 3) = 0 → u = 2 or u = 3
Recover x: 2ˣ = 2 → x = 1; 2ˣ = 3 → x = ln(3)/ln(2)
x = 1 or x ≈ 1.585
Natural Log — Base e Equations
When the base is e, take the natural log of both sides. Because ln(eˣ) = x, the exponential and log cancel cleanly.
Solve: e²ˣ = 7
Take ln of both sides: ln(e²ˣ) = ln(7)
Simplify: 2x = ln(7)
x = ln(7) / 2 ≈ 0.973
Solving Logarithmic Equations — 3 Types
A logarithmic equation has a variable inside a logarithm. After solving, always verify that your answer keeps every log argument positive.
Always check your domain
Log equations can produce extraneous solutions — values that satisfy the algebra but make a log argument zero or negative. These must be discarded. The argument of every log in the original equation must be strictly positive.
Single Logarithm — Convert to Exponential Form
Isolate the log on one side, then rewrite the equation in exponential form to solve for x.
Solve: log₂(x) = 5
Rewrite in exponential form: 2⁵ = x
x = 32 ✓ (argument 32 > 0)
Solve: 3 · log(x) − 2 = 4
Isolate the log: 3 · log(x) = 6 → log(x) = 2
Exponential form: 10² = x
x = 100 ✓
Multiple Logs — Combine Using Laws First
When the equation has multiple logs with the same base, combine them into one log using product, quotient, or power rules, then convert to exponential form.
Solve: log(x) + log(x + 3) = 1
Product rule: log(x(x + 3)) = 1
Exponential form: x(x + 3) = 10¹
Expand: x² + 3x − 10 = 0
Factor: (x + 5)(x − 2) = 0 → x = −5 or x = 2
Check: x = −5 makes log(−5) undefined → discard
x = 2 ✓ (both arguments 2 and 5 are positive)
Log = Log — Set Arguments Equal
When a single log expression with the same base appears on both sides, drop the logs and set the arguments equal. This works because log is a one-to-one function.
Solve: log₃(2x − 1) = log₃(x + 4)
Same base → set arguments equal: 2x − 1 = x + 4
Solve: x = 5
Check: argument 1 → 2(5) − 1 = 9 > 0 ✓; argument 2 → 5 + 4 = 9 > 0 ✓
x = 5 ✓
Graphs of Exponential and Logarithmic Functions
Exponential and logarithmic functions are inverses of each other. Their graphs are reflections over the line y = x. Understanding key features lets you sketch either graph quickly.
y = bˣ (b > 1)
- Domain: all real numbers (−∞, ∞)
- Range: (0, ∞) — always positive
- y-intercept: (0, 1) — because b⁰ = 1
- No x-intercept — curve never touches x-axis
- Horizontal asymptote: y = 0 (as x → −∞)
- Behavior: increasing, grows rapidly for large x
y = log_b(x) (b > 1)
- Domain: (0, ∞) — x must be positive
- Range: all real numbers (−∞, ∞)
- x-intercept: (1, 0) — because log_b(1) = 0
- No y-intercept — curve never touches y-axis
- Vertical asymptote: x = 0 (the y-axis)
- Behavior: increasing, grows very slowly
Inverse Functions — Reflection Over y = x
Because y = bˣ and y = log_b(x) are inverse functions, each undoes the other:
b^(log_b(x)) = x (for x > 0)
log_b(bˣ) = x (for all x)
ln(eˣ) = x
e^(ln x) = x (for x > 0)
On a graph, if you fold along the line y = x, the curve y = bˣ lands exactly on y = log_b(x). The coordinates swap: if (2, 8) is on y = 2ˣ, then (8, 2) is on y = log₂(x).
Real-World Applications — 3 Worked Problems
Compound Interest — Solve for Time
Continuously compounded interest uses the formula A = P · e^(rt), where A is the final amount, P is the principal, r is the annual rate, and t is time in years.
Problem: $1,000 is invested at 5% annual rate, continuously compounded. When does it reach $2,000?
A = P · e^(rt) → 2000 = 1000 · e^(0.05t)
Divide both sides by 1000: 2 = e^(0.05t)
Take ln of both sides: ln(2) = 0.05t
Solve: t = ln(2) / 0.05
t ≈ 13.86 years
Key step: isolate the exponential first, then take ln of both sides. The rule of 70 (70/r%) gives a quick estimate for doubling time.
Radioactive Half-Life
Radioactive decay follows A = A₀ · (1/2)^(t/h), where A₀ is the initial amount, h is the half-life, and t is time.
Problem: A substance has a half-life of 12 years. How long until only 25% remains?
0.25 · A₀ = A₀ · (1/2)^(t/12)
Divide by A₀: 0.25 = (1/2)^(t/12)
Note: 0.25 = (1/2)² → same base!
(1/2)² = (1/2)^(t/12) → t/12 = 2
t = 24 years
Alternative — using ln:
Take ln: ln(0.25) = (t/12) · ln(1/2)
t = 12 · ln(0.25) / ln(0.5) = 12 · (−1.386)/(−0.693)
t = 24 years ✓
pH and Hydrogen Ion Concentration
The pH scale is defined by pH = −log[H⁺], where [H⁺] is the hydrogen ion concentration in moles per liter (mol/L).
Problem: A solution has pH = 3.5. Find the hydrogen ion concentration [H⁺].
pH = −log[H⁺] → 3.5 = −log[H⁺]
Multiply both sides by −1: log[H⁺] = −3.5
Convert to exponential form: [H⁺] = 10^(−3.5)
[H⁺] ≈ 3.16 × 10⁻⁴ mol/L
Each pH unit represents a factor of 10 change in [H⁺]. pH 3 is 10× more acidic than pH 4, and 100× more acidic than pH 5.
Frequently Asked Questions
What is a logarithm?
A logarithm is an exponent. The expression log_b(x) = y means exactly the same thing as b^y = x — it asks 'to what power must I raise b to get x?' For example, log_2(8) = 3 because 2^3 = 8. The three most common bases are base 10 (written log), base e (written ln), and base 2 (used in computer science). Understanding that 'the logarithm IS the exponent' is the key to every log problem.
What are the logarithm laws and how do you use them?
There are six logarithm laws: (1) Product rule: log(ab) = log a + log b — multiplication inside becomes addition outside. (2) Quotient rule: log(a/b) = log a − log b — division inside becomes subtraction outside. (3) Power rule: log(a^n) = n·log a — an exponent inside comes out front as a multiplier. (4) Change of base: log_b(x) = ln(x)/ln(b) — converts any base to natural log or log base 10. (5) log_b(b) = 1 — a log of its own base is always 1. (6) log_b(1) = 0 — a log of 1 is always 0 because b^0 = 1.
How do you solve logarithmic equations?
There are three main types. (1) Single log equation: isolate the log, then convert to exponential form. For log_2(x) = 5, rewrite as 2^5 = x, so x = 32. (2) Multiple log equation with the same base: use log laws to combine logs into one, then convert to exponential form. For log(x) + log(x+3) = 1, combine to log(x(x+3)) = 1, then x(x+3) = 10, and solve the resulting equation. (3) Log equals log: if log_b(A) = log_b(B) then A = B — set the arguments equal and solve. In every case, you MUST check your answers — if any solution makes the argument of a log zero or negative, discard it.
Practice logarithm and exponential problems
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