Decimal Logarithm: Definition and Properties

The decimal logarithm, also known as the common logarithm, is a fundamental mathematical concept with wide-ranging applications. This page introduces the definition of the decimal logarithm and outlines its key properties.

Definition: The decimal logarithm of a positive number b, denoted as log(b), is the exponent to which 10 must be raised to obtain b. In other words, if 10^x = b, then x = log(b).

The logarithm function is defined for all positive real numbers, making its domain (0, +∞). This function is the inverse of the exponential function with base 10.

Highlight: The equation 10^x = b (where b > 0) has a unique solution, which is denoted as log(b). This is the fundamental principle of the decimal logarithm.

Some key properties of the decimal logarithm include:

1. log(1) = 0, as 10^0 = 1
2. log(10) = 1, as 10^1 = 10
3. For any positive real number x, 10^(log(x)) = x
4. For any real number y, log(10^y) = y

Example: To solve the equation 10^x = 1000, we can apply the logarithm to both sides: log(10^x) = log(1000). Using the properties of logarithms, this simplifies to x = log(1000) = 3.

The logarithm function has several important properties that make it useful in various mathematical and practical applications:

1. log(ab) = log(a) + log(b)
2. log(a/b) = log(a) - log(b)
3. log(a^n) = n * log(a)

Vocabulary: The fonction logarithme décimal (decimal logarithm function) is the function defined on (0, +∞) that associates x with log(x).

Understanding these properties and the nature of the logarithm function is crucial for solving more complex problems involving exponentials and logarithms in fields such as science, engineering, and finance.

Highlight: For all positive real numbers a and b, log(a) = log(b) if and only if a = b. This property is essential for solving equations involving logarithms.

The decimal logarithm function is strictly increasing on its domain, which means:

• If x > 0, then log(x) < 0
• If x > 1, then log(x) > 0

This property is particularly useful when comparing numbers or solving inequalities involving logarithms.