Common Logarithm

Definition

The common logarithm, also known as the base-10 logarithm, is a logarithmic function that expresses the power to which a base of 10 must be raised to obtain a given number. It is a fundamental concept in mathematics that is widely used in various fields, including science, engineering, and finance.

5 Must Know Facts For Your Next Test

1. The common logarithm of a number x is denoted as $\log_{10}(x)$, which represents the power to which 10 must be raised to obtain x.
2. The common logarithm of 1 is 0, the common logarithm of 10 is 1, and the common logarithm of 100 is 2.
3. Common logarithms are used to represent very large or very small numbers in a more compact and manageable form, such as in scientific notation.
4. The properties of common logarithms, including the power rule, product rule, and quotient rule, are essential in solving exponential and logarithmic equations.
5. Common logarithms are closely related to the concept of exponential functions, as they represent the inverse operation of exponentiation with a base of 10.

Review Questions

• Explain how common logarithms are used to represent and manipulate very large or very small numbers.
  • Common logarithms are used to represent very large or very small numbers in a more compact and manageable form, such as in scientific notation. For example, the number 1,000,000 can be expressed as $10^6$, where the common logarithm of 1,000,000 is 6. Similarly, the number 0.000001 can be expressed as $10^{-6}$, where the common logarithm of 0.000001 is -6. This allows for easier manipulation and calculation of these extreme values.
• Describe the relationship between common logarithms and exponential functions, and how this relationship is utilized in solving exponential and logarithmic equations.
  • Common logarithms are closely related to the concept of exponential functions, as they represent the inverse operation of exponentiation with a base of 10. This inverse relationship is fundamental in solving exponential and logarithmic equations. For example, if we have the equation $2^x = 32$, we can solve for $x$ by taking the common logarithm of both sides: $\log_{10}(2^x) = \log_{10}(32)$, which simplifies to $x \log_{10}(2) = \log_{10}(32)$, and finally $x = \log_{10}(32) / \log_{10}(2)$. This demonstrates how common logarithms are used to convert between exponential and logarithmic forms.
• Analyze how the properties of common logarithms, such as the power rule, product rule, and quotient rule, are applied in the context of solving exponential and logarithmic equations.
  • The properties of common logarithms, including the power rule, product rule, and quotient rule, are essential in solving exponential and logarithmic equations. For instance, the power rule states that $\log_{10}(x^n) = n\log_{10}(x)$, which can be used to simplify expressions involving exponents. The product rule states that $\log_{10}(xy) = \log_{10}(x) + \log_{10}(y)$, allowing for the manipulation of logarithmic expressions involving multiplication. Similarly, the quotient rule states that $\log_{10}(x/y) = \log_{10}(x) - \log_{10}(y)$, which is useful for simplifying logarithmic expressions involving division. By understanding and applying these properties, one can effectively solve a wide range of exponential and logarithmic equations.

Related terms

Logarithmic Function: A logarithmic function is a function in which the independent variable appears as an exponent. It is the inverse of an exponential function.

Natural Logarithm: The natural logarithm, or base-e logarithm, is a logarithmic function that uses the mathematical constant e (approximately 2.718) as the base.

Exponential Function: An exponential function is a function in which the dependent variable is raised to a power that is determined by the independent variable.