key term - Log

Definition

A logarithm is the exponent to which a base number must be raised to get a certain value. It represents the power to which a base number must be raised to produce a given number. Logarithms are used to simplify complex calculations and are fundamental to understanding logarithmic functions and their properties.

5 Must Know Facts For Your Next Test

1. Logarithms allow us to represent very large or very small numbers in a more compact form, which is useful for scientific calculations.
2. The common logarithm, with base 10, is denoted as $\log_10(x)$ or simply $\log(x)$, and represents the power to which 10 must be raised to get the value $x$.
3. The natural logarithm, with base $e$ (approximately 2.718), is denoted as $\ln(x)$ and represents the power to which $e$ must be raised to get the value $x$.
4. Logarithmic functions are the inverse of exponential functions, meaning that $\log_b(x) = y$ if and only if $b^y = x$.
5. Logarithmic properties, such as the product rule, power rule, and logarithm of a quotient, allow for simplification and manipulation of logarithmic expressions.

Review Questions

• Explain how logarithmic functions are related to exponential functions.
  • Logarithmic functions and exponential functions are inverse functions of each other. This means that if $y = b^x$, then $x = \log_b(y)$. The logarithm represents the power to which the base $b$ must be raised to get the value $y$. For example, if $y = 10^3$, then $x = \log_10(1000) = 3$, since $10^3 = 1000$. This inverse relationship allows logarithms to be used to simplify complex exponential calculations.
• Describe the key properties of logarithmic functions that are useful for solving problems.
  • The main logarithmic properties that are important for problem-solving include the product rule ($\log_b(xy) = \log_b(x) + \log_b(y)$), the power rule ($\log_b(x^n) = n\log_b(x)$), and the logarithm of a quotient ($\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$). These properties allow for the manipulation and simplification of logarithmic expressions, which is essential for solving a variety of mathematical problems involving exponential and logarithmic functions.
• Explain how the graphs of logarithmic functions differ from the graphs of exponential functions, and how this relationship can be used to analyze the behavior of these functions.
  • The graph of a logarithmic function, such as $y = \log_b(x)$, is the inverse of the graph of the corresponding exponential function, $y = b^x$. While exponential functions have a characteristic J-shaped curve that increases rapidly, logarithmic functions have a concave-down shape that increases more slowly and approaches the horizontal axis asymptotically. This inverse relationship between the graphs of logarithmic and exponential functions can be used to analyze the behavior of these functions, such as identifying key features like domain, range, and transformations, as well as to solve problems involving exponential growth and decay.

Related terms

Base:

The number that is used as the reference point for a logarithm. Common logarithmic bases are 10 (common logarithm) and e (natural logarithm).

Exponential Function:

A function where the variable appears in the exponent, such as $y = b^x$, where $b$ is the base. Logarithmic functions are the inverse of exponential functions.

Inverse Function:

A function that undoes the operation of another function. Logarithmic functions are the inverse of exponential functions, and vice versa.