Understanding Logarithms and Their Properties

Definition: A logarithm is a mathematical operator whose main objective is to transform very large or very small numbers into a more manageable scale. The logarithm of a number a with base b is denoted as log_b a and is defined by the equation:

log_b a = x if and only if b^x = a.

Important Note: The base b is always a positive real number different from one.

Graphical Representation

We will work on this in the second trimester when studying logarithmic functions.

Properties of Logarithms

1. Product: The logarithm of a product is the sum of the logarithms.

   log_b(x·y) = log_b x + log_b y

2. Quotient: The logarithm of a quotient is the difference of the logarithms.

   log_b(x/y) = log_b x - log_b y

3. Power: The logarithm of a power is the exponent times the logarithm of the base.

   log_b(xⁿ) = n·log_b x

4. Root: The logarithm of a root is the logarithm of the base divided by the root's degree.

   log_b(ⁿ√x) = (1/n)·log_b x

Other Interesting Properties

• Logarithms of negative numbers are undefined.
• The logarithm of a number less than one is always negative.
• log_b 1 = 0 because any number raised to the power of zero is one.
• log_b b = 1 because any number raised to the power of one is itself.

Equations and Systems of Logarithmic Equations

These are always solved by applying the properties of logarithms to transform the original expressions into equivalent ones where logarithmic operators do not appear.