The common logarithm, also known as the base-10 logarithm, is a logarithmic function that expresses how many times a number must be multiplied by 10 to get a given number. It is a fundamental concept in the study of logarithmic functions and their properties.
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The common logarithm of a number $x$ is denoted as $\log_{10}(x)$ or $\log(x)$, and it represents the power to which 10 must be raised to get the value $x$.
The common logarithm of 1 is 0, as $10^0 = 1$. The common logarithm of 10 is 1, as $10^1 = 10$.
Common logarithms are useful for simplifying calculations, as they allow for the conversion of multiplication and division operations into addition and subtraction.
The properties of logarithms, such as the product rule, quotient rule, and power rule, are essential for evaluating and manipulating logarithmic expressions.
Graphing common logarithmic functions reveals their characteristic concave-up shape, which is useful for understanding the behavior of logarithmic functions.
Review Questions
Explain how the common logarithm is defined and how it relates to the base-10 number system.
The common logarithm, denoted as $\log(x)$ or $\log_{10}(x)$, is a logarithmic function that expresses the power to which 10 must be raised to get a given number $x$. In other words, if $10^y = x$, then $\log(x) = y$. This relationship between the common logarithm and the base-10 number system makes common logarithms particularly useful for simplifying calculations involving multiplication and division.
Describe how the properties of logarithms, such as the product rule, quotient rule, and power rule, can be applied to evaluate and manipulate logarithmic expressions.
The properties of logarithms allow for the conversion of multiplication and division operations into addition and subtraction. For example, the product rule states that $\log(a \cdot b) = \log(a) + \log(b)$, the quotient rule states that $\log(a \div b) = \log(a) - \log(b)$, and the power rule states that $\log(a^n) = n \cdot \log(a)$. These properties are essential for simplifying and evaluating logarithmic expressions, which is a crucial skill in the context of using and applying logarithmic functions.
Explain how the graph of a common logarithmic function, $y = \log(x)$, can be used to understand the behavior of logarithmic functions.
The graph of a common logarithmic function, $y = \log(x)$, has a characteristic concave-up shape. This shape reflects the fact that as the input $x$ increases, the rate of change of the output $y$ decreases. This behavior is important for understanding the properties of logarithmic functions, such as their asymptotic approach to the $x$-axis and their ability to transform multiplicative relationships into additive ones. Analyzing the graph of a common logarithmic function can provide valuable insights into the behavior and applications of logarithmic functions in various contexts.