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Precalculus: Exponential and Logarithmic Equations & Compound Interest

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Terms in this set (20)

  • What is an exponential equation?
    An equation containing a variable in an exponent.
  • How do you solve exponential equations using the One-to-One Property?
    Rewrite each side with the same base, then set the exponents equal.
  • What steps are used to solve exponential equations using logarithms?
    1) Isolate the exponential term. 2) Take log of both sides. 3) Use log properties to simplify. 4) Solve for the variable.
  • What is the formula for compound interest compounded n times per year?
    A = P \(\left(1 + \frac{r}{n}\right)^{nt}\), where A is accumulated value, P is principal, r is annual rate, n is compounding frequency, t is time in years.
  • How do you solve logarithmic equations with logs on one side?
    Rewrite as log_b(x) = c, then convert to exponential form b^c = x and solve.
  • What must be checked after solving logarithmic equations?
    Check that solutions are in the domain of the original logarithmic expressions (arguments > 0).
  • How do you solve logarithmic equations with logs on both sides?
    Set the logs equal, use the one-to-one property to equate arguments, then solve.
  • What is the natural logarithm used for in solving exponential equations?
    It is used to solve equations where the base cannot be rewritten to be the same, especially with base e.
  • How do you find the time to double an investment with compound interest?
    Use the formula A = P \(\left(1 + \frac{r}{n}\right)^{nt}\), set A = 2P, then solve for t using logarithms.
  • What is the domain restriction for logarithmic functions?
    The argument of a logarithm must be greater than zero.
  • How do you express the equation log_3 x = 4 in exponential form?
    Rewrite as 3^4 = x, so x = 81.
  • What is the solution to the equation 2^x = 16 using the One-to-One Property?
    Rewrite 16 as 2^4, so x = 4.
  • How do you solve e^{3x} = 8 using logarithms?
    Take natural log: 3x = ln 8, then x = (ln 8)/3.
  • What is the solution to log_2 (x + 5) = 3?
    Rewrite as 2^3 = x + 5, so x = 8 - 5 = 3.
  • How do you solve the logarithmic equation log_5(x) + log_5(x - 4) = 1?
    Combine logs: log_5[x(x - 4)] = 1, then 5^1 = x^2 - 4x, solve quadratic.
  • What is the formula to find the accumulated amount with continuous compounding?
    A = P \(e^{rt}\), where e is Euler's number.
  • How do you solve for x in the equation 10^x = 3.91?
    Take log base 10: x = log 3.91.
  • What is the solution to 2 log x = log 25?
    Rewrite as log x^2 = log 25, so x^2 = 25, x = ±5, but domain restricts x > 0, so x = 5.
  • How do you solve the equation log_3 x = log_3 5 + log_3 (x - 2)?
    Combine right side: log_3 [5(x - 2)] = log_3 x, so x = 5(x - 2), solve for x.
  • How do you calculate the time t for an investment to grow to a certain amount with monthly compounding?
    Use A = P \(\left(1 + \frac{r}{12}\right)^{12t}\), solve for t using logarithms.