What is a Logarithm?

A logarithm is a mathematical function that determines the exponent or power to which a base number must be raised to produce a given number. In other words, if you have an equation of the form:

ax = b

then the logarithm of b to the base a is x, written as:

loga(b) = x

Example:

If 23 = 8, then log2(8) = 3.

Properties of Logarithms:

• Product Rule: loga(xy) = loga(x) + loga(y)
• Quotient Rule: loga(x/y) = loga(x) - loga(y)
• Power Rule: loga(xn) = n × loga(x)
• Change of Base Formula: loga(b) = logc(b) / logc(a), where c is any positive number different from 1.

Common Bases:

• Base 10: log10(x) is called the common logarithm.
• Base e: loge(x) is called the natural logarithm and is denoted as ln(x), where e (Euler's number) is approximately 2.718.

Logarithms are widely used in various fields like science, engineering, and finance to simplify multiplication and division operations, solve exponential equations, and model exponential growth or decay.