Primer on Logarithms

Logarithms are the "inverse of exponentiation." If you’ve ever wondered “what power of a number produces another number?”, you’re working with logarithms. This primer introduces logarithms, explains their properties, and proves those properties step by step.

What is a Logarithm?

The logarithm answers the question:
"To what power must the base be raised to produce a given number?"

Mathematically:

• : the base of the logarithm (e.g., 2, 10, or ).
• : the number you’re taking the logarithm of.
• : the exponent.

Example:

because .

Key Properties of Logarithms

Logarithms simplify complex operations. Below are the fundamental properties, along with proofs.

1. Product Rule

Proof:

1. Let and . By the definition of logarithms:
2. The product is:
3. By the laws of exponents, , so:
4. Take of both sides:
5. By the property , we get:
6. Substitute back and :

2. Quotient Rule

Proof:

1. Let and . By the definition of logarithms:
2. The quotient is:
3. By the laws of exponents, , so:
4. Take of both sides:
5. By the property , we get:
6. Substitute back and :

3. Power Rule

Proof:

1. Let . By the definition of logarithms:
2. Raise both sides to the power :
3. By the laws of exponents, , so:
4. Take of both sides:
5. By the property , we get:
6. Substitute back :

4. Change of Base Formula

Proof:

1. Let . By the definition of logarithms:
2. Take of both sides (using a new base ):
3. By the power rule, :
4. Solve for :
5. Recall that , so:

Applications of Logarithms

Logarithms have widespread applications:

• Growth and Decay: Population dynamics, radioactive decay, and compound interest.
• Scales: Decibels (sound), pH (acidity), and earthquake magnitudes.
• Computing: Algorithm efficiency (e.g., ), binary trees, and compression.

Quick Practice Problems

1. Prove .
2. Simplify .
3. Solve for : .

Logarithms are incredibly powerful and essential across mathematics, science, and engineering. Mastering their properties will make complex calculations and analysis much simpler!