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Introduction to Logarithmic Functions quiz

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  • What is a logarithm in relation to an exponential function?
    A logarithm is the exponent that a base must be raised to in order to equal a particular number; it is the inverse of an exponential function.
  • How do you write the exponential equation 2^y = 10 in logarithmic form?
    You write it as y = log base 2 of 10, or y = log₂(10).
  • What does the notation log_b(x) mean?
    It means the exponent y such that b^y = x; b is the base and x is the argument.
  • How are exponential and logarithmic forms related?
    They are inverses: y = log_b(x) is equivalent to x = b^y.
  • What is the base in the logarithmic expression log_5(x)?
    The base is 5.
  • How do you convert the logarithmic equation log_3(81) = 4 to exponential form?
    You write it as 3^4 = 81.
  • What is the inverse function of f(x) = 5^x?
    The inverse is f⁻¹(x) = log base 5 of x, or f⁻¹(x) = log₅(x).
  • What is the value of log_2(2^3)?
    It is 3, because the log and exponential with the same base cancel, leaving the exponent.
  • What is the value of log_b(b) for any base b?
    It is always 1, because b^1 = b.
  • What is the value of log_b(1) for any base b?
    It is always 0, because any base raised to the power of 0 equals 1.
  • How can you evaluate log_4(16) without a calculator?
    Rewrite 16 as 4^2, so log_4(4^2) = 2.
  • How do you evaluate log_5(1/5) using exponent rules?
    Rewrite 1/5 as 5^(-1), so log_5(5^(-1)) = -1.
  • How do you rewrite the cube root of 4 as an exponent?
    The cube root of 4 is 4^(1/3).
  • What is log_4(∛4) equal to?
    It is 1/3, because log_4(4^(1/3)) = 1/3.
  • What two elements should you focus on when converting between exponential and logarithmic forms?
    Focus on the base and the exponent, as they determine the structure of both forms.