Precalc
Section 4.3

Logarithmic Functions

Logs show up on every exam that covers Chapter 4 — expect 2–3 questions just on evaluating and converting logs, plus they're the main tool for solving exponential equations in Sections 4.5–4.7. Get comfortable converting between log and exponential form and the rest of the chapter clicks.

⚡ Quick Summary

A logarithm is just a fancy way of asking: "what power do I need?" For example, log⁡2(8)\log_2(8) asks "2 to what power gives me 8?" The answer is 3, because 23=82^3 = 8. That's all a log does — it finds the missing exponent. If you can think of every log as an exponent question, the notation stops being scary.

Definition of logarithm
log⁡b(x)=y  ⟺  by=x\log_b(x) = y \iff b^y = x

b > 0, b ≠ 1, and x > 0

Common logarithm
log⁡(x)=log⁡10(x)\log(x) = \log_{10}(x)

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

A logarithm is just a fancy way of asking: "what power do I need?" For example, log⁡2(8)\log_2(8) asks "2 to what power gives me 8?" The answer is 3, because 23=82^3 = 8. That's all a log does — it finds the missing exponent. If you can think of every log as an exponent question, the notation stops being scary.

You'll see two special logs everywhere: **log⁡\log (which secretly means base 10) and ln⁡\ln** (which means base e≈2.718e ≈ 2.718). Your calculator has buttons for both. For any other base, there's a simple trick called the change-of-base formula — you just divide two ln⁡\ln values to convert: log⁡b(x)=ln⁡xln⁡b\log_b(x) = \frac{\ln x}{\ln b}.

Logs and exponentials are like undo buttons for each other. If exponentials are the question "what do I get when I raise bb to this power?", then logs are the reverse: "what power do I need to get this number?" This is why log⁡b(bx)=x\log_b(b^x) = x — the log peels off the exponent — and blog⁡b(x)=xb^{\log_b(x)} = x — the exponential rebuilds the number.

Logs show up in real life more than you'd think. The Richter scale (earthquakes), decibels (sound), and pH (chemistry) are all logarithmic — they compress huge ranges of numbers into a manageable scale. A magnitude 7 earthquake is 10 times more powerful than a magnitude 6, not just one unit more. That compression is the whole point of logarithms.

Key Formulas

Definition of logarithm
log⁡b(x)=y  ⟺  by=x\log_b(x) = y \iff b^y = x

b > 0, b ≠ 1, and x > 0

Common logarithm
log⁡(x)=log⁡10(x)\log(x) = \log_{10}(x)
Natural logarithm
ln⁡(x)=log⁡e(x)\ln(x) = \log_e(x)

e ≈ 2.718

Inverse relationships
log⁡b(bx)=xandblog⁡b(x)=x\log_b(b^x) = x \quad \text{and} \quad b^{\log_b(x)} = x

Logs and exponentials cancel each other

Change-of-base formula
log⁡b(x)=ln⁡(x)ln⁡(b)=log⁡(x)log⁡(b)\log_b(x) = \frac{\ln(x)}{\ln(b)} = \frac{\log(x)}{\log(b)}

Use this to evaluate any log on a calculator

Key Takeaways

  • ✓Logarithms answer the question: what exponent gives me this number? log⁡b(x)=y\log_b(x) = y means by=xb^y = x
  • ✓The natural log ln⁡\ln uses base e≈2.718e \approx 2.718; the common log log⁡\log uses base 10
  • ✓You can only take the log of a positive number — the domain of log⁡b(x)\log_b(x) is x>0x > 0
  • ✓Use the change-of-base formula log⁡b(x)=ln⁡xln⁡b\log_b(x) = \frac{\ln x}{\ln b} to evaluate any log on a calculator

⚠️ Common Mistakes

  • ✗Thinking log⁡(x+y)=log⁡(x)+log⁡(y)\log(x + y) = \log(x) + \log(y) — there is NO log rule for sums; the product rule is log⁡(xy)=log⁡x+log⁡y\log(xy) = \log x + \log y
  • ✗Forgetting that you can only take the log of a positive number — log⁡b(0)\log_b(0) and log⁡b(−5)\log_b(-5) are undefined
  • ✗Confusing log⁡\log (base 10) with ln⁡\ln (base ee) — when no base is written, it's base 10
  • ✗Getting the conversion backward: log⁡b(x)=y\log_b(x) = y means by=xb^y = x, not bx=yb^x = y
Next: Section 4.4Builds on: 4.1
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4.2 Graphs of Exponential Functions
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4.4 Graphs of Logarithmic Functions