Log Functions- Properties and Graphing Guide

What Log Functions Actually Are

Log functions are the inverse operations of exponential functions. If you have by = x, the logarithm answers: what exponent y gives you x when the base is b?

You write it as logb(x) = y. That's it. No magic, no mystery.

The most common bases you'll see are:

• Base 10 — written as log(x) in most textbooks
• Base e (natural log) — written as ln(x)
• Base 2 — written as log2(x), common in computer science

You'll use these constantly in calculus, algebra, and any field that deals with growth or decay. Know them cold.

The Five Properties You Must Memorize

These aren't suggestions. These are the rules that govern every log problem you'll encounter.

Product Rule

logb(MN) = logb(M) + logb(N)

When multiplying inside a log, you split it into addition outside. Works every time.

Quotient Rule

logb(M/N) = logb(M) - logb(N)

Division inside becomes subtraction outside. Same pattern as the product rule, just flipped.

Power Rule

logb(MN) = N · logb(M)

The exponent comes down and multiplies. This one saves you from expanding ridiculous expressions.

Change of Base Formula

logb(x) = loga(x) / loga(b)

When your calculator only has base 10 or base e, you use this to convert. No calculator gives you base 7 directly — this formula fixes that.

Inverse Property

logb(bx) = x and blogb(x) = x

Logs and exponentials cancel each other out. This is how you solve equations — you undo one operation with its inverse.

Domain and Range: What You Can't Do

Log functions have strict rules about what numbers they accept. Ignore these and you'll get burned.

Domain restriction: The input must be positive. log(x) doesn't exist for x ≤ 0. No negative numbers, no zero. Ever.

Range: Log functions can output any real number — positive, negative, zero. There's no ceiling or floor.

This means the graph approaches the y-axis (vertical asymptote) but never touches it. The curve lives entirely to the right of x = 0.

Graphing Log Functions: The Practical Method

You don't need to plot 50 points. You need three anchor points and an understanding of the shape.

Step 1: Identify the Base and Any Transformations

For y = logb(x - h) + k:

• h shifts the graph horizontally (opposite sign)
• k shifts the graph vertically (same sign)

Step 2: Find the Key Points

Every log graph passes through (1, 0) because logb(1) = 0 for any base.

For base > 1 (increasing), also plot:

• (b, 1) — the base point
• (1/b, -1) — the reciprocal point

Step 3: Draw the Asymptote

The vertical asymptote is at x = h (after accounting for horizontal shift). Draw a dashed line here — the curve approaches it but never crosses.

Step 4: Connect the Dots

The graph is always increasing for base > 1, always decreasing for base between 0 and 1. No wiggles, no turns. Just a smooth curve.

Common Mistakes That Cost Points

• Forgetting the domain. Students lose marks constantly because they include x ≤ 0 in their solution sets. Don't be that person.
• Mixing up product and quotient rules. Multiplication inside = addition outside. Division inside = subtraction outside. Write it on your hand if you have to.
• Applying power rule incorrectly. The exponent multiplies the whole log, not just the base. log(x2) = 2log(x), not log(x2) = log(2x).
• Confusing ln and log. In higher math, log usually means natural log. In algebra, it usually means base 10. Check the context — or ask.

Solving Log Equations: A Worked Example

Problem: Solve log3(x + 5) = 4

Step 1: Rewrite in exponential form. Log equations are just disguised exponentials.

34 = x + 5

Step 2: Solve the resulting equation.

81 = x + 5

x = 76

Step 3: Check your work. The domain requires x + 5 > 0, so x > -5. 76 passes, so you're good.

Solving Systems of Log Equations

When you have multiple log expressions, use the properties to combine or expand until you isolate the variable.

Problem: Solve log2(x) + log2(x - 2) = 3

Use the product rule in reverse: combine the left side.

log2(x(x - 2)) = 3

log2(x2 - 2x) = 3

Convert to exponential: x2 - 2x = 23

x2 - 2x = 8

x2 - 2x - 8 = 0

(x - 4)(x + 2) = 0

x = 4 or x = -2

Domain check: x > 0 and x > 2 (from the second log). Only x = 4 works. Reject -2.

Log Functions vs. Exponential Functions: The Comparison

They mirror each other across the line y = x. Whatever shape one makes, the other reflects it perfectly.

Applications: Where Logs Actually Show Up

Logs aren't abstract torture devices. They measure real phenomena:

• Richter scale — earthquake magnitude uses log₁₀
• pH scale — acidity measured with negative logs
• Sound (decibels) — uses log₁₀ for pressure ratios
• Compound interest — continuous interest involves ln
• Information theory — entropy uses log₂

The math you learn here connects directly to chemistry, physics, engineering, and data science. It's not wasted time.

Getting Started: Your Action Plan

If you're learning this for a class, here's what actually works:

1. Memorize the five properties — write them out 10 times until they're automatic
2. Practice converting between log and exponential form until it's second nature
3. Graph by hand at least five different log functions with various transformations
4. Solve 20 equations — start simple, add complexity
5. Check every answer against the domain restrictions

You don't need expensive tutors or fancy software. You need repetition. Log functions click once you put in the reps.

Bottom Line

Log functions are inverses of exponentials. They have strict domain rules. The properties let you expand, condense, and solve equations. The graphs are predictable once you know the anchor points and asymptote location.

No memorization hack replaces understanding why the rules work. But if you just need to pass the test — memorize the five properties, practice the conversions, and always check your domain.