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.