ResourcesMathematicsExponents & Logarithms
MathematicsHigh School

Exponents & Logarithms

Exponents represent repeated multiplication, and logarithms are their inverse. Together they form a fundamental pair of operations used throughout science, engineering, finance, and computer science.

This guide covers key definitions, the laws of exponents and logarithms, exponential and logarithmic functions, solving techniques, worked examples, common mistakes, and a practice quiz.

1Introduction

At its core, exponentiation is repeated multiplication. For example, 2³ means 2 × 2 × 2 = 8. Logarithms reverse this process: if exponentiation asks "What is b raised to the power x?", a logarithm asks "To what power must b be raised to get y?"

These two operations are inverses of each other, and understanding both is essential for mastering higher mathematics.

The Inverse Relationship

If 2³ = 8, then log₂ 8 = 3. Exponents and logarithms are two sides of the same coin — one builds up, the other unwinds.

Why Do They Matter?

Science

Population growth, radioactive decay, Richter scale (earthquakes), pH scale (chemistry), sound intensity (decibels).

Finance

Compound interest, investment growth, loan repayments, and the Rule of 72.

Computer Science

Algorithm complexity (O(log n)), binary search, data structures, information theory.

Engineering

Signal processing, control systems, exponential smoothing, and electrical circuits.

2Key Definitions

Base (b)

The number being multiplied repeatedly. In bˣ, b is the base. Example: in 2³, the base is 2.

Exponent / Power (x)

Indicates how many times the base is multiplied by itself. In 2³, 3 is the exponent.

Logarithm (log₂ y)

The exponent to which the base must be raised to produce y. If 2³ = 8, then log₂ 8 = 3.

Common Logarithm (log x)

Logarithm with base 10. Written as log x or log₁₀ x. Example: log 100 = 2.

Natural Logarithm (ln x)

Logarithm with base e (≈ 2.71828). Written as ln x. Example: ln e⁵ = 5.

Antilogarithm

The inverse of a logarithm. If log₁₀ x = 2, then x = 10² = 100.

The core relationship

bˣ = y ⇔ log₂ y = x

Exponential form and logarithmic form express the same fact.

3Laws of Exponents

These rules let you simplify and manipulate expressions involving exponents. Let a and b be real numbers, and m and n be rational numbers.

Exponent Rules Explorer

aᵐ · aⁿ = aᵐ⁺ⁿ

Same base? Add the exponents.

Example: x³ · x⁵ = x⁸

1. Product Rule: aᵐ · aⁿ = aᵐ⁺ⁿ

x³ · x⁵ = x³⁺⁵ = x⁸

2⁴ · 2⁻¹ = 2³ = 8

2. Quotient Rule: aᵐ / aⁿ = aᵐ⁻ⁿ

y⁷ / y² = y⁵

5³ / 5⁵ = 5⁻² = 1/25

3. Power Rule: (aᵐ)ⁿ = aᵐⁿ

(z⁴)³ = z¹²

(2²)³ = 2⁶ = 64

4. Product to a Power: (ab)ⁿ = aⁿbⁿ

(2x)³ = 2³ · x³ = 8x³

5. Quotient to a Power: (a/b)ⁿ = aⁿ/bⁿ

(x/3)² = x²/3² = x²/9

6. Zero Exponent: a⁰ = 1

5⁰ = 1

(where a ≠ 0)

7. Negative Exponent: a⁻ⁿ = 1/aⁿ

4⁻² = 1/4² = 1/16

8. Fractional Exponent: aᵐ˸ⁿ = ⁿ√(aᵐ)

x½ = √x

8²˳ = (³√8)² = 2² = 4

4Exponential Functions

An exponential function has the form f(x) = abˣ, where a is the initial value, b is the base (b > 0, b ≠ 1), and x is the exponent.

f(x) = abˣ

a = initial value (y-intercept when x = 0), b = base (growth or decay factor).

Key Characteristics

Y-intercept

Always (0, a). If a = 1, the graph passes through (0, 1).

Horizontal Asymptote

The x-axis (y = 0). The graph approaches but never touches it.

Domain

All real numbers (−∞, ∞).

Range

All positive real numbers (0, ∞) when a > 0.

Growth vs. Decay

Exponential Growth (b > 1)

y = 2ˣ

As x increases, y increases rapidly. Used for population growth and compound interest.

Exponential Decay (0 < b < 1)

y = (0.5)ˣ

As x increases, y decreases toward zero. Used for radioactive decay and depreciation.

5Introduction to Logarithms

Logarithms answer the question: "To what power must the base be raised to get a certain number?" They are the inverse operation of exponentiation.

Converting Between Forms

2³ = 8 ⇔ log₂ 8 = 3

10² = 100 ⇔ log₁₀ 100 = 2

5⁰ = 1 ⇔ log₅ 1 = 0

3⁻² = 1/9 ⇔ log₃ (1/9) = −2

Logarithm Calculator

Enter a base and argument to compute the logarithm. Try different values to build intuition!

Special Logarithms

Common Logarithm

log x = log₁₀ x

Base 10. Used in scientific scales, engineering, and everyday calculations.

Natural Logarithm

ln x = logₑ x

Base e (≈ 2.718). Preferred in calculus and higher mathematics.

Logarithmic Function Characteristics

X-intercept

Always (1, 0) because log₂ 1 = 0 for any base.

Vertical Asymptote

The y-axis (x = 0). The graph approaches but never crosses it.

Domain

x > 0 only. The argument must be positive.

Range

All real numbers (−∞, ∞).

6Laws of Logarithms

These laws are derived directly from the laws of exponents and are essential for simplifying logarithmic expressions and solving equations.

1. Product Rule: log₂(MN) = log₂ M + log₂ N

log₂(4 · 8) = log₂ 4 + log₂ 8

= 2 + 3 = 5

Check: log₂ 32 = 5 ✓

2. Quotient Rule: log₂(M/N) = log₂ M − log₂ N

log₃(81/3) = log₃ 81 − log₃ 3

= 4 − 1 = 3

Check: log₃ 27 = 3 ✓

3. Power Rule: log₂(Mᵖ) = p · log₂ M

log₅(25³) = 3 · log₅ 25

= 3 · 2 = 6

4. Change of Base: log₂ M = log M / log b = ln M / ln b

log₂ 10 = log(10) / log(2)

≈ 1 / 0.301 ≈ 3.32

Important Identities

log₂ b = 1
log₂ 1 = 0
blog₂ x = x
log₂(bˣ) = x

7Solving Exponential Equations

An exponential equation has the variable in the exponent. There are two main methods for solving them.

Method 1: Making Bases the Same

If you can rewrite both sides with the same base, equate the exponents: if bᵐ = bⁿ, then m = n.

Solve: 2ˣ⁺¹ = 8

Rewrite 8 as 2³:

2ˣ⁺¹ = 2³

x + 1 = 3

x = 2

Method 2: Taking Logarithms

When bases cannot be easily matched, take the log of both sides and use the power rule.

Solve: 3ˣ = 20

log(3ˣ) = log(20)

x · log(3) = log(20) (power rule)

x = log(20) / log(3)

x ≈ 2.727

Solve: 5²ˣ⁻¹ = 125

Rewrite 125 as 5³:

5²ˣ⁻¹ = 5³

2x − 1 = 3

2x = 4

x = 2

8Solving Logarithmic Equations

A logarithmic equation involves logarithms of expressions containing the variable. Always check for extraneous solutions — the argument of a logarithm must be positive.

Method 1: Convert to Exponential Form

Solve: log₂(x − 3) = 4

Convert: 2⁴ = x − 3

16 = x − 3

x = 19

Check: log₂(19 − 3) = log₂ 16 = 4 ✓

Solve: log₃ x + log₃(x − 8) = 2

log₃(x(x − 8)) = 2 (product rule)

3² = x(x − 8)

9 = x² − 8x

x² − 8x − 9 = 0

(x − 9)(x + 1) = 0

x = 9 or x = −1

x = −1: log₃(−1) is undefined — reject

x = 9

Method 2: One-to-One Property

If log₂ M = log₂ N, then M = N (same base, equate arguments).

Solve: ln(x + 1) − ln(x − 1) = ln 3

ln((x + 1)/(x − 1)) = ln 3 (quotient rule)

(x + 1)/(x − 1) = 3

x + 1 = 3(x − 1)

x + 1 = 3x − 3

4 = 2x

x = 2

Check: ln 3 − ln 1 = ln 3 ✓

Critical Reminder

Always check for extraneous solutions! Substitute your answers back in and verify that every logarithm argument is positive. Solutions that make any argument zero or negative must be rejected.

9Worked Examples

Example 1: Simplify using exponent rules

Simplify: (3x²y⁻³)² · (2x⁻¹y⁴)

= (9x⁴y⁻⁶) · (2x⁻¹y⁴) (power rule)

= 18 · x⁴⁺⁻¹ · y⁻⁶⁺⁴ (product rule)

= 18x³y⁻²

= 18x³/y²

Example 2: Evaluate without a calculator

Evaluate: log₄ 64 + log₃(1/9)

log₄ 64: 4³ = 64, so log₄ 64 = 3

log₃(1/9): 3⁻² = 1/9, so log₃(1/9) = −2

3 + (−2) = 1

Example 3: Solve an exponential equation

Solve: 5²ˣ⁻¹ = 125

125 = 5³

5²ˣ⁻¹ = 5³

2x − 1 = 3

2x = 4

x = 2

Example 4: Solve a logarithmic equation

Solve: ln(x + 1) − ln(x − 1) = ln 3

ln((x+1)/(x−1)) = ln 3 (quotient rule)

(x+1)/(x−1) = 3

x + 1 = 3x − 3

4 = 2x

x = 2

Check: ln 3 − ln 1 = ln 3 − 0 = ln 3 ✓

Example 5: Real-world application (exponential decay)

A car bought for

0,000 depreciates at 15% per year. Write a model and find its value after 3 years.

V(t) = P(1 − r)ᵗ

V(t) = 20000(1 − 0.15)ᵗ

V(t) = 20000(0.85)ᵗ

V(3) = 20000(0.85)³

V(3) = 20000(0.614125)

V(3) =

2,282.50

10Common Mistakes

Confusing negative exponents with negative numbers

2⁻³ ≠ −8. A negative exponent means reciprocal: 2⁻³ = 1/2³ = 1/8.

Distributing exponents over addition

(a + b)ⁿ ≠ aⁿ + bⁿ. For example, (x + y)² ≠ x² + y². The power rule only works for products: (ab)ⁿ = aⁿbⁿ.

log(A + B) ≠ log A + log B

There is no logarithm rule for a sum. The product rule says log(AB) = log A + log B, but this only applies to multiplication, not addition.

Confusing log A / log B with log(A/B)

log A / log B is the change of base formula. The quotient rule is log A − log B = log(A/B). These are different operations.

Forgetting logarithm domain restrictions

The argument of a logarithm must be positive. Always check solutions: log(x − 5) requires x > 5. Reject extraneous solutions.

Order of operations with negative signs

−2² = −(2²) = −4, but (−2)² = (−2)(−2) = 4. The exponent only applies to the base it is directly attached to.

Pro Tip

When in doubt, convert back to exponential form. The relationship bˣ = y is the anchor that connects exponents and logarithms. If you get stuck on a log problem, rewrite it as an exponent to check your reasoning.

Quick Revision Summary

  • ✓Exponents represent repeated multiplication: bˣ. Logarithms are the inverse: log₂ y = x means bˣ = y.
  • ✓Laws of exponents: product (add), quotient (subtract), power (multiply), zero (= 1), negative (reciprocal), fractional (root).
  • ✓Exponential functions f(x) = abˣ model growth (b > 1) or decay (0 < b < 1) with horizontal asymptote y = 0.
  • ✓Laws of logarithms: product rule (add logs), quotient rule (subtract logs), power rule (exponent out front), change of base.
  • ✓Common log (log) has base 10. Natural log (ln) has base e ≈ 2.718.
  • ✓Solving exponential equations: match bases or take logarithms of both sides.
  • ✓Solving logarithmic equations: condense logs, convert to exponential form, or use the one-to-one property.
  • ✓Always check for extraneous solutions — logarithm arguments must be positive.

Frequently Asked Questions

What's the difference between x² and 2ˣ?
x² is a polynomial (quadratic) function where the base is the variable and the exponent is a constant. Its graph is a parabola. 2ˣ is an exponential function where the base is a constant and the exponent is the variable. Its graph shows rapid growth. They behave very differently as x increases.
Why can't the base of a logarithm be 1?
If b = 1, then 1 raised to any power is always 1. This means log₁(y) could be any number when y = 1, making it not a unique function, and it would be undefined for any y ≠ 1. To ensure a well-defined inverse function, the base must not be 1.
Why can't the argument of a logarithm be negative or zero?
Because the base b of a logarithm must be positive (b > 0), raising a positive number to any real power always gives a positive result. Since log_b(y) = x means bˣ = y, the value y must always be positive. You can never get zero or a negative number by raising a positive base to a real power.
When should I use log (common log) vs ln (natural log)?
Common log (log₁₀) is used for scales like the Richter scale or pH, and when dealing with powers of 10. Natural log (ln) is preferred in calculus and higher mathematics because of properties of e that simplify many formulas. For solving equations, either works, but ln is more common in advanced math.
How do I remember the logarithm rules?
Think of them as inverses of exponent rules: multiplication of powers (add exponents) corresponds to the log product rule (add logs). Division of powers (subtract exponents) corresponds to the log quotient rule (subtract logs). Power-to-a-power (multiply exponents) corresponds to the log power rule (exponent comes out front).

Practice Quiz

Test your understanding of exponents and logarithms — select the correct answer for each question.

1.What is x³ × x²?

2.What is 4⁰?

3.What is log₁₀(100)?

4.What is log₂(1)?

5.Simplify: log₃(9) + log₃(3)

6.What is e to the power of ln(x)?

7.Solve: 3ˣ = 27

8.What is log₂(8)?

9.Simplify: 10 to the power of log(x)

10.What is the base of the natural logarithm (ln)?

Final Study Advice

  • 1.Memorize the laws of exponents first — the log laws are their direct inverses.
  • 2.When stuck on a log problem, convert to exponential form to see the relationship more clearly.
  • 3.Practice converting between exponential and logarithmic forms until it becomes automatic.
  • 4.Always check your solutions in logarithmic equations for extraneous values.
  • 5.Use the change of base formula to evaluate unusual logarithms on your calculator.

Related Topics