Log

Definition

A logarithm is the exponent to which a base number must be raised to get another number. It represents the power to which a fixed number, the base, must be raised to produce a given number.

5 Must Know Facts For Your Next Test

1. Logarithmic functions are the inverse of exponential functions, allowing us to solve exponential equations.
2. The graph of a logarithmic function is concave down and approaches the x-axis asymptotically.
3. The properties of logarithms, such as the product rule, power rule, and logarithm of a quotient, allow us to simplify and evaluate logarithmic expressions.
4. Logarithmic equations can be solved by rewriting them as exponential equations and then applying the properties of logarithms.
5. Logarithms are used in various fields, including physics, chemistry, and computer science, to represent and manipulate data that spans a wide range of values.

Review Questions

• Explain how logarithmic functions are related to exponential functions and how this relationship is used to evaluate and graph logarithmic functions.
  • Logarithmic functions are the inverse of exponential functions. This means that if $y = a^x$, then $x = extbackslash log_a y$. This inverse relationship allows us to evaluate logarithmic functions by rewriting them as exponential equations and solving for the input value. Additionally, the graph of a logarithmic function is the reflection of the graph of the corresponding exponential function about the line $y = x$, which is a useful property for graphing logarithmic functions.
• Describe the properties of logarithms and explain how they are used to simplify and evaluate logarithmic expressions.
  • The key properties of logarithms include the product rule ($ extbackslash log_a (xy) = extbackslash log_a x + extbackslash log_a y$), the power rule ($ extbackslash log_a (x^n) = n extbackslash log_a x$), and the logarithm of a quotient ($ extbackslash log_a (x/y) = extbackslash log_a x - extbackslash log_a y$). These properties allow us to manipulate logarithmic expressions by rewriting them in simpler forms, which is essential for evaluating and solving problems involving logarithms.
• Discuss how logarithmic equations can be solved and explain the importance of logarithms in various fields of study.
  • Logarithmic equations can be solved by rewriting them as exponential equations and then applying the properties of logarithms. This process involves isolating the logarithmic term and then using the inverse relationship between logarithmic and exponential functions to find the solution. Logarithms are widely used in fields such as physics, chemistry, and computer science because they allow for the representation and manipulation of data that spans a wide range of values, making calculations and data analysis more efficient and practical.

Related terms

Exponential Function: A function of the form $f(x) = a^x$, where $a$ is a positive real number called the base.

Inverse Function: A function that 'undoes' the operation of another function, such that $f(f^{-1}(x)) = x$.

Common Logarithm: A logarithm with base 10, denoted as $ extbackslash log x$.