Logarithms

Definition

Logarithms are a mathematical function that represents the power to which a base number must be raised to get another number. They are used to express very large or very small numbers in a more compact form and are closely related to exponential functions.

5 Must Know Facts For Your Next Test

1. Logarithms are the inverse operation of exponents, allowing us to solve for the power or exponent when given the result.
2. The logarithm of a number $x$ with base $b$ is the power to which $b$ must be raised to get $x$, written as $ ext{log}_b(x)$.
3. Logarithms can be used to solve exponential equations by converting them to linear equations.
4. Common logarithms use a base of 10, while natural logarithms use a base of $e \approx 2.718$.
5. Logarithms have important applications in fields like physics, chemistry, biology, and computer science, where they are used to represent and analyze data.

Review Questions

• Explain how logarithms are related to exponential functions and how they can be used to solve exponential equations.
  • Logarithms and exponential functions are inverse operations. The logarithm of a number $x$ with base $b$ is the power to which $b$ must be raised to get $x$, written as $ ext{log}_b(x)$. This means that if $y = b^x$, then $x = ext{log}_b(y)$. By converting an exponential equation to a logarithmic equation, we can solve for the unknown exponent or power. For example, to solve the equation $2^x = 32$, we can take the logarithm of both sides: $ ext{log}_2(2^x) = ext{log}_2(32)$, which simplifies to $x = ext{log}_2(32) = 5$.
• Describe the difference between common logarithms (base 10) and natural logarithms (base $e$), and explain when each type of logarithm might be more appropriate to use.
  • Common logarithms, which use a base of 10, are often used in scientific and engineering applications where data is frequently represented in powers of 10. Natural logarithms, which use a base of $e \approx 2.718$, are more commonly used in mathematical analysis and in fields like economics and finance, where exponential growth and decay are important. The choice of which logarithm to use depends on the context and the specific problem being solved. Common logarithms are more intuitive for working with large numbers, while natural logarithms have useful mathematical properties that make them advantageous in certain types of analysis.
• Analyze the role of logarithms in the study of exponential functions and their applications in various scientific and technological fields.
  • Logarithms are fundamental to the study of exponential functions, as they represent the inverse operation. This allows us to solve exponential equations, model exponential growth and decay, and analyze data represented in exponential form. Logarithms have widespread applications in fields like physics (e.g., decibel scale for sound intensity), chemistry (e.g., pH scale), biology (e.g., population growth models), and computer science (e.g., logarithmic time complexity of algorithms). The ability to represent large or small numbers using logarithms, as well as the mathematical properties of logarithms, make them an essential tool for scientific and technological advancement. Understanding the relationship between logarithms and exponential functions is crucial for interpreting and working with data in these diverse domains.

Related terms

Exponential Function: A function in which the variable appears in the exponent, such as $y = b^x$, where $b$ is the base of the exponential function.

Base: The number that is raised to a power in an exponential function or logarithm. Common bases include 10, 2, and $e$ (the natural logarithm).

Power: The exponent to which a base number is raised in an exponential function or logarithm.