📘intermediate algebra review

key term - Logarithm

Definition

A logarithm is a mathematical function that represents the power to which a fixed number, called the base, must be raised to obtain a given number. Logarithms are closely related to exponential functions and are used to simplify complex calculations, especially in the context of growth and decay processes.

5 Must Know Facts For Your Next Test

  1. Logarithms are used to evaluate and graph exponential functions, as they represent the exponents to which a base must be raised to obtain a given value.
  2. The logarithmic function is the inverse of the exponential function, allowing for the solution of exponential equations.
  3. The properties of logarithms, such as the product rule, quotient rule, and power rule, are essential for simplifying and manipulating logarithmic expressions.
  4. Logarithms are used to solve exponential and logarithmic equations by converting them to linear equations.
  5. The natural logarithm, with a base of $e$, is particularly important in scientific and mathematical applications.

Review Questions

  • Explain how logarithms are used to evaluate and graph exponential functions.
    • Logarithms are used to evaluate and graph exponential functions because they represent the exponents to which a base must be raised to obtain a given value. The logarithm of a number is the exponent to which the base must be raised to get that number. For example, if $2^3 = 8$, then the logarithm of 8 with base 2 is 3, written as $\log_2 8 = 3$. This relationship between logarithms and exponents allows us to transform exponential functions into linear functions, which can then be easily evaluated and graphed.
  • Describe how the properties of logarithms, such as the product rule, quotient rule, and power rule, are used to simplify and manipulate logarithmic expressions.
    • The properties of logarithms, such as the product rule ($\log_b (xy) = \log_b x + \log_b y$), the quotient rule ($\log_b (x/y) = \log_b x - \log_b y$), and the power rule ($\log_b (x^n) = n\log_b x$), allow for the simplification and manipulation of logarithmic expressions. These properties enable us to rewrite complex logarithmic expressions in a more manageable form, which is particularly useful when solving exponential and logarithmic equations or simplifying expressions involving logarithms.
  • Analyze how logarithms are used to solve exponential and logarithmic equations.
    • Logarithms are used to solve exponential and logarithmic equations by converting them to linear equations. For example, to solve an exponential equation like $2^x = 32$, we can take the logarithm of both sides to obtain $x = \log_2 32$, which is a linear equation. Similarly, to solve a logarithmic equation like $\log_3 x = 5$, we can rewrite it as $x = 3^5$, which is an exponential equation. By utilizing the inverse relationship between logarithms and exponential functions, we can transform these types of equations into a more manageable form and solve for the unknown variable.