Lesson 5.9

Common and Natural Logs

Two bases are used so often they get their own buttons on the calculator: (base 10) and (base ). Most applied problems use one of these.

Introduction

Writing every time is tedious, so we shorten it to just . Similarly, is written (from the Latin logarithmus naturalis). These are the only two logs your calculator can compute directly.

Past Knowledge

Log definition (5.6), evaluating logs (5.8), the number (5.3).

Today's Goal

Use and notation and evaluate with a calculator.

Future Success

Solving exponential equations (5.16) requires or on the calculator.

Key Concepts

Common Log:

Base 10 — the "LOG" button on your calculator.

Quick facts:

Natural Log:

Base — the "LN" button.

Key inverse pairs:

Change of Base Formula

This lets you evaluate any base log on a calculator that only has LOG and LN.

Worked Examples

Example 1: Common Log

Basic

Evaluate .

1

Calculator: LOG → 500 → ENTER

Makes sense: 500 is between and

Example 2: Natural Log

Basic

Evaluate .

1

Calculator: LN → 10 → ENTER

Since

Example 3: Change of Base

Intermediate

Evaluate .

1

Change of base

Check: ✓

Common Pitfalls

Confusing LOG and LN

= base 10. = base . They give different values! but .

Change of Base Fraction Upside-Down

It's , not . The argument goes on top, the base on the bottom.

Real-Life Applications

The pH scale uses . The decibel scale uses . Scientists choose for modeling because simplifies calculus, while engineers prefer for scales.

Practice Quiz

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