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Precalculus: Exponential and Logarithmic Functions

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Terms in this set (20)

  • What is the general form of an exponential function?
    An exponential function has the form \(f(x) = b^x\) where b is a positive constant, b > 0 and b \(\neq\) 1, and x is the independent variable.
  • What is the natural base e and its approximate value?
    e is an irrational number approximately equal to 2.71828, discovered by Bernoulli in 1683, and is the base of the natural exponential function.
  • What is the natural exponential function?
    The natural exponential function is \(f(x) = e^x\), where e is Euler's constant.
  • What are the domain and range of exponential functions with base b > 1?
    Domain: \((-\infty, \infty)\), Range: \((0, \infty)\). The graph passes through (0,1) and has a horizontal asymptote at y=0.
  • What are the domain and range of exponential functions with 0 < b < 1?
    Domain: \((-\infty, \infty)\), Range: \((0, \infty)\). The graph passes through (0,1) and has a horizontal asymptote at y=0, but the function is decreasing.
  • What is the formula for compound interest compounded n times per year?
    \(A = P \left(1 + \frac{r}{n}\right)^{nt}\), where A is the amount, P is the principal, r is the annual interest rate, n is the number of compounding periods per year, and t is time in years.
  • What is the formula for continuously compounded interest?
    \(A = Pe^{rt}\), where A is the amount, P is the principal, r is the annual interest rate, and t is time in years.
  • What is the inverse function of an exponential function \(f(x) = b^x\)?
    The inverse is the logarithmic function \(f^{-1}(x) = \log_b x\), which answers the question: to what power must b be raised to get x?
  • What is the domain of a logarithmic function \(f(x) = \log_b x\)?
    The domain is all positive real numbers: \((0, \infty)\). The argument of the log must be positive.
  • Write the logarithmic equation \(\log_b x = y\) in exponential form.
    The equivalent exponential form is \(b^y = x\).
  • Write the exponential equation \(b^y = x\) in logarithmic form.
    The equivalent logarithmic form is \(\log_b x = y\).
  • What is the value of \(\log_b 1\) for any base b?
    \(\log_b 1 = 0\) because \(b^0 = 1\).
  • What is the value of \(\log_b b\) for any base b?
    \(\log_b b = 1\) because \(b^1 = b\).
  • What is the common logarithm?
    The common logarithm is the logarithm with base 10, written as \(\log x\) or \(\log_{10} x\).
  • What is the natural logarithm?
    The natural logarithm is the logarithm with base e, written as \(\ln x\).
  • Evaluate \(\log_4 16\) without a calculator.
    Since \(4^2 = 16\), \(\log_4 16 = 2\).
  • Evaluate \(\log_{10} 100\) without a calculator.
    Since \(10^2 = 100\), \(\log_{10} 100 = 2\).
  • Evaluate \(\log_2 1/4\) without a calculator.
    Since \(2^{-2} = 1/4\), \(\log_2 \frac{1}{4} = -2\).
  • Find the domain of \(f(x) = \log_2 (x + 9)\).
    Set the argument > 0: \(x + 9 > 0 \Rightarrow x > -9\). Domain: \((-9, \infty)\).
  • Find the domain of \(f(x) = \ln (5 - x)\).
    Set the argument > 0: \(5 - x > 0 \Rightarrow x < 5\). Domain: \((-\infty, 5)\).