Logarithm

Definition

A logarithm is a mathematical function that describes the power to which a base number must be raised to get a certain value. It represents the exponent to which a base number must be raised to produce a given number. Logarithms are closely related to exponential functions and are essential in understanding topics such as logarithmic functions, graphs of logarithmic functions, exponential and logarithmic equations, and geometric sequences.

5 Must Know Facts For Your Next Test

1. Logarithms are used to represent very large or very small numbers in a compact form, making calculations more manageable.
2. The common logarithm (base 10) is denoted as $\log x$, while the natural logarithm (base $e$) is denoted as $\ln x$.
3. Logarithmic functions are used to model growth and decay processes, such as the growth of populations and the decay of radioactive materials.
4. Graphs of logarithmic functions are concave down and asymptotic to the x-axis, reflecting the logarithm's slow growth rate.
5. Solving exponential and logarithmic equations often involves converting between the two forms using the properties of logarithms.

Review Questions

• Explain how logarithms are related to exponential functions and how this relationship is used in the study of logarithmic functions.
  • Logarithms and exponential functions are inverse operations. An exponential function $y = b^x$ can be expressed as a logarithmic function $x = \log_b y$, where $b$ is the base of the logarithm. This inverse relationship is crucial in the study of logarithmic functions, as it allows us to convert between exponential and logarithmic forms and analyze the properties and graphs of logarithmic functions. For example, the concave-down shape of logarithmic functions and their asymptotic behavior can be understood by examining the inverse relationship with exponential functions.
• Describe how logarithms are used in the context of solving exponential and logarithmic equations.
  • Logarithms are essential in solving exponential and logarithmic equations. When faced with an exponential equation, such as $2^x = 16$, we can apply the logarithm to both sides to convert it to a logarithmic equation: $x = \log_2 16$. Similarly, when solving a logarithmic equation, such as $\log_3 x = 5$, we can apply the inverse operation of exponentiation to both sides to convert it to an exponential equation: $x = 3^5$. This interplay between exponential and logarithmic forms is a crucial skill in solving a variety of equations involving these functions.
• Explain how the properties of logarithms, such as the power rule and the logarithm of a product, can be applied in the context of geometric sequences.
  • The properties of logarithms are particularly useful in the study of geometric sequences, which exhibit exponential growth or decay. For example, the formula for the $n$th term of a geometric sequence, $a_n = a_1 \cdot r^{n-1}$, can be rewritten using logarithms as $\log_r \left(\frac{a_n}{a_1}\right) = n-1$. This allows us to apply logarithm properties, such as the power rule $\log_b (x^y) = y\log_b(x)$, to analyze and solve problems involving the terms and behavior of geometric sequences. The relationship between logarithms and exponential functions is fundamental in understanding the underlying patterns and properties of geometric sequences.

Related terms

Base: The base of a logarithm is the number that is raised to a power to produce a given value. Common logarithm bases include 10 and the natural logarithm base e.

Exponential Function: An exponential function is the inverse of a logarithmic function, where a variable is raised to a power rather than the power being the unknown.

Inverse Function: A function that undoes the operation of another function. Logarithms are the inverse functions of exponential functions.