Logarithm Function- Properties, Graphs, and Applications

What Is a Logarithm?

A logarithm answers a simple question: what exponent produces a given number?

If 2³ = 8, then log₂(8) = 3. That's it. The logarithm tells you the power you need when the base is fixed.

Mathematicians write it as log_b(x) = y, which means b^y = x. The base is b, the result is x, and y is the exponent you're solving for.

Types of Logarithms

Common Logarithm (log)

Base 10. When you see log(x) without a subscript, assume base 10. Useful for decibel scales and scientific notation.

Natural Logarithm (ln)

Base e (approximately 2.718). Written as ln(x). Shows up constantly in calculus, growth models, and probability. You can't simplify e into a rational number, so natural logs are unavoidable in advanced math.

Binary Logarithm (log₂)

Base 2. Used in computer science for algorithm analysis and data storage calculations.

Core Properties of Logarithms

These rules let you manipulate logs algebraically. Commit these to memory—they appear in every logarithm problem you'll encounter.

• Product Rule: log_b(xy) = log_b(x) + log_b(y)
• Quotient Rule: log_b(x/y) = log_b(x) − log_b(y)
• Power Rule: log_b(xⁿ) = n · log_b(x)
• Change of Base: log_b(x) = log_k(x) / log_k(b) for any positive base k
• Zero Property: log_b(1) = 0 (because b⁰ = 1)
• Identity Property: log_b(b) = 1

Why the Change of Base Formula Matters

Calculators only give you log₁₀ and ln. If you need log₂(50), convert it: log₂(50) = ln(50)/ln(2). Both ln values are calculator-accessible.

Graphing Logarithmic Functions

The graph of y = log_b(x) has distinct features you need to recognize.

Key Characteristics

• Domain: x > 0 only. You cannot take a log of zero or a negative number.
• Range: All real numbers (−∞, ∞)
• Y-intercept: None. The graph never crosses the y-axis.
• X-intercept: (1, 0). Every log function passes through this point.
• Vertical Asymptote: x = 0. The graph approaches but never touches the y-axis.

How the Base Affects the Shape

When b > 1, the function increases as x increases. When 0 < b < 1, the function decreases as x increases—the graph flips horizontally.

📈 For b > 1: steeper curve as b increases
📉 For 0 < b < 1: curve falls and approaches the asymptote from above

Inverse Relationship with Exponentials

Log and exponential functions are inverses. The graph of y = log_b(x) is a reflection of y = b^x across the line y = x. If you understand exponentials, logs make intuitive sense.

Logarithm Function Comparison

Real-World Applications

Logs aren't abstract exercises. They measure things that grow or shrink exponentially.

Richter Scale

Each whole number increase represents 10 times more ground motion. An earthquake measuring 6.0 has 10× the amplitude of a 5.0 earthquake. The formula: M = log₁₀(A/A₀).

pH Scale

pH = −log₁₀[H⁺]. Neutral water has pH 7. Battery acid (pH 1) has 1,000,000× more hydrogen ions than neutral water.

Compound Interest

Money growing continuously follows A = Pe^rt. To find how long it takes to double, solve 2 = e^rt, which gives t = ln(2)/r. This is why natural logs appear in finance.

Sound Measurement (Decibels)

dB = 10 × log₁₀(I/I₀). A 70 dB sound is 10× more intense than 60 dB. Your ears perceive this logarithmically, which is why decibels use logs.

Algorithm Analysis

Binary search runs in O(log₂n) time. If you have 1 billion items, binary search takes at most log₂(10⁹) ≈ 30 comparisons. Without logs, you'd need 1 billion comparisons in a linear search.

Solving Logarithmic Equations

Here's how to actually work with logs in practice.

Example 1: Basic Evaluation

Solve log₃(81) = ?

Ask: 3 to what power gives 81?
3⁴ = 81
Answer: 4

Example 2: Converting Between Forms

Rewrite 2⁵ = 32 as a logarithm.

log₂(32) = 5. The base becomes the base of the log, the result becomes the argument, and the exponent becomes the answer.

Example 3: Using Log Properties

Simplify log₂(8) + log₂(4).

Option A (product rule): log₂(8 × 4) = log₂(32) = 5

Option B (evaluate separately): log₂(8) = 3, log₂(4) = 2. Sum = 5

Example 4: Solving for x

Solve 2^x = 17.

Take log of both sides: log(2^x) = log(17)
Apply power rule: x · log(2) = log(17)
Solve: x = log(17)/log(2)
x ≈ 4.09

Example 5: Change of Base

Calculate log₅(100) using a standard calculator.

log₅(100) = ln(100)/ln(5)
= 4.605/1.609
≈ 2.86

Common Mistakes to Avoid

• log(x + y) ≠ log(x) + log(y). This is wrong. Only products split into sums.
• log(x)/log(y) ≠ log_x(y). You need to swap: log_x(y) = log(y)/log(x).
• Domain restrictions are real. If your solution gives log(−3), it's invalid. Check your work.
• Confusing ln and log₁₀. In higher math, log usually means ln. In science and engineering, it often means base 10. Check context.

When to Use Logarithms

Reach for logs when you're dealing with exponential relationships and need to:

• Isolate an exponent in an equation
• Compress large ranges (like earthquake magnitudes or sound levels)
• Transform multiplicative relationships into additive ones
• Find doubling times or half-lives

Logs turn multiplication into addition, powers into multiplication, and exponentials into linear functions. That's the real value—they make complicated growth manageable.