Logarithmic Expressions

A logarithm answers a deceptively simple question: 'To what power must I raise a given base to obtain a certain number?' If you know that 2 raised to the 5th power equals 32, then the logarithm base 2 of 32 is 5. Written symbolically, log₂(32) = 5. Logarithms are the inverse operation of exponentiation, just as subtraction undoes addition and division undoes multiplication. This inverse relationship is the key to evaluating and manipulating logarithmic expressions. Logarithms appear throughout mathematics, science, and everyday life. The Richter scale for earthquakes, the decibel scale for sound, the pH scale for acidity, and compound interest calculations all rely on logarithms. Because exponential growth and decay are everywhere in nature, the ability to 'undo' an exponential — which is exactly what a logarithm does — is an essential mathematical tool. Two bases appear so frequently that they have special names: log base 10 is called the common logarithm (written simply as log), and log base e ≈ 2.71828 is called the natural logarithm (written as ln). To work confidently with logarithmic expressions, you need to master three skills: converting between exponential and logarithmic form, evaluating logarithms exactly or approximately, and applying the three fundamental logarithm laws — the product rule, quotient rule, and power rule. These laws let you expand complicated logarithmic expressions into simpler pieces, or condense multiple logarithms into a single expression, both of which are critical for solving logarithmic equations and simplifying calculus computations later in your studies.

Key takeaways
  • The Definition: Switching Between Forms
  • Special Values Every Student Must Know
  • The Product Rule
  • The Quotient Rule
  • The Power Rule
  • The Change-of-Base Formula

The Definition: Switching Between Forms

The fundamental definition states: log_b(x) = y ⟺ b^y = x, where b > 0, b ≠ 1, and x > 0. The base b is the same in both forms; the exponent y is what the logarithm outputs; and x is the result of the exponentiation. Practice switching: log₃(81) = 4 because 3⁴ = 81. Conversely, 5³ = 125 means log₅(125) = 3. The domain of log_b(x) is x > 0 — you cannot take the log of a negative number or zero.

Special Values Every Student Must Know

Three results follow directly from the definition and should be memorized. (1) log_b(1) = 0 for any valid base, because b⁰ = 1. (2) log_b(b) = 1 for any valid base, because b¹ = b. (3) log_b(b^n) = n, because that exponent n is exactly what the logarithm recovers. The inverse relationship also gives b^(log_b(x)) = x. For example: log(10,000) = log(10⁴) = 4; ln(e³) = 3; log₅(1) = 0; log₇(7) = 1.

The Product Rule

When the argument of a logarithm is a product, it splits into a sum: log_b(M · N) = log_b(M) + log_b(N). This mirrors the exponent rule b^m · b^n = b^(m+n). Example: log₂(8 · 4) = log₂(8) + log₂(4) = 3 + 2 = 5. Check: log₂(32) = 5 ✓. You can also use this in reverse to combine separate logarithms with the same base into one.

The Quotient Rule

When the argument is a quotient, the logarithm splits into a difference: log_b(M/N) = log_b(M) − log_b(N). This mirrors b^m / b^n = b^(m−n). Example: log₃(243/9) = log₃(243) − log₃(9) = 5 − 2 = 3. Check: 243/9 = 27 = 3³, so log₃(27) = 3 ✓. Important: log_b(M) − log_b(N) ≠ log_b(M − N). The subtraction happens outside, not inside.

The Power Rule

When the argument has an exponent, the exponent moves out front as a coefficient: log_b(M^p) = p · log_b(M). This is perhaps the most useful rule for solving equations, because it brings a variable down from an exponent. Example: log₂(64) = log₂(2⁶) = 6 · log₂(2) = 6 · 1 = 6. Another: log(√x) = log(x^(1/2)) = (1/2)log(x). The rule also applies to negative exponents: log(1/x) = log(x^(−1)) = −log(x).

The Change-of-Base Formula

Most calculators only compute log (base 10) and ln (base e). To evaluate any other base, use the change-of-base formula: log_b(x) = log(x)/log(b) = ln(x)/ln(b). Example: log₅(200) = log(200)/log(5) = 2.3010/0.6990 ≈ 3.292. You can verify: 5^3.292 ≈ 200 ✓. This formula also proves that all logarithmic functions of different bases are just scalar multiples of each other.

The Key Equation

log⁡b(x)=y  ⟺  by=x(b>0,  b≠1,  x>0)\log_b(x) = y \iff b^y = x \qquad (b > 0,\; b \neq 1,\; x > 0)
  • b — Base (dimensionless): The fixed positive number (not equal to 1) that is repeatedly multiplied; must satisfy b > 0 and b ≠ 1.
  • x — Argument (dimensionless): The positive input to the logarithm; the number whose logarithm is being found. Must be strictly positive.
  • y — Logarithm value (exponent) (dimensionless): The output of the logarithm; the exponent to which b must be raised to produce x. Can be any real number.

Worked Example

Problem. Expand the expression log₂( (8x⁵) / √y ) into simpler logarithms, then evaluate the constant part exactly.

3+5log⁡2(x)−12log⁡2(y)3 + 5\log_2(x) - \tfrac{1}{2}\log_2(y)

Answer. 3 + 5·log₂(x) − (1/2)·log₂(y)

Figure
Diagram
Graph of Logarithmic Functionsxy01234-112(1, 0)(1, 0)(3, log₂3≈1.58) waity = log₂(x)y = log₃(x)Both curves pass through (1, 0) and increase; larger base = slower growth
Origins · 1 minJohn Napier and how this was discovered
JN
John Napier
1550–1617 · Scottish

John Napier invented logarithms and published his landmark work 'Mirifici Logarithmorum Canonis Descriptio' (A Description of the Marvellous Rule of Logarithms) in 1614. His tables allowed multiplication and division of large numbers to be replaced by addition and subtraction — a revolutionary labor-saving tool for astronomers and navigators. He also invented 'Napier's bones,' a manual calculating device. The natural logarithm base e is sometimes called the Napierian logarithm in his honor, though Napier's original construction used a slightly different base. Napier was so interested in apocalyptic theology that he wrote a lengthy commentary on the Book of Revelation. His neighbors reportedly believed he practiced witchcraft, partly because he wandered his estate at night with a black rooster he claimed was his familiar spirit — but historians believe this was a clever act to make his tenants fear cheating him.

  • Exponential Functions — Logarithms are the inverse of exponential functions; understanding y = bˣ is prerequisite to understanding log_b(x).
  • Properties of Logarithms — The product, quotient, and power rules introduced here are explored in greater depth to solve complex logarithmic equations.
  • Solving Logarithmic Equations — Evaluating and expanding logarithmic expressions is the core skill needed to isolate variables in logarithmic equations.
  • The Natural Logarithm — The natural logarithm ln(x) is a special case of log_b(x) with base e, central to calculus and continuous growth models.
  • Inverse Functions — Logarithms are formally the inverse functions of exponentials; the concept of function inverses underpins the entire logarithm definition.