Exponential & Log Equations

  1. For each of the following equations, find the value(s) of \(x\) that make the equation true. If \(x\) is not a “nice” number, round it to three decimal places.
    \( \displaystyle 2^{3x+7} = 16 \)
    \( \displaystyle 4\Bigl(3^{x-7}\Bigr) = 972 \)
    \( \displaystyle 42 = \mathrm{e}^{x^2+1} \)
  2. For each of the following equations, find the value(s) of \(x\) that make the equation true. If \(x\) is not a “nice” number, round it to three decimal places.
    \(\displaystyle \log_{10}(14x-1) = 3 \)
    \(\displaystyle 7=\ln\bigl(x^2\bigr) \)
    \(\displaystyle \log_3\bigl(7+\log_5(x+3)\bigr) = 2 \)
  3. A tribble is an alien creature that, given an adequate food supply, is known to produce a litter of ten offspring every twelve hours. Due to this fact, if you start with a population of \(N_0\) tribbles, after \(t\) twelve-hour intervals the size of the population will be modelled by the function \(N(t) = N_0 (11)^t.\)
    1. If you have an initial population of only two tribbles, how long before there are one million tribbles?
    2. Rewrite the equation for \(N(t)\) as an exponential function with base \(\mathrm{e},\) the natural base. I.e., you’ll need to find \(k\) such that \(N_0 (11)^t = N_0\mathrm{e}^{kt}.\)
  4. The formula for the future value \(A\) of an initial investment of \(P\) dollars invested at an annual interest rate of \(r\) percent compounded annually for \(t\) years is given by the formula \[A= P(1+r)^t.\]
    1. If you deposit $420 in a savings account that gets \(3\%\) annual interest how long before the account balance is $500? (Note that \(3\% = 0.03\,.\))
    2. A more realistic annual interest rate for a savings account at one of the Big Four US banks, like Chase of Bank of America, is \(0.01\%\). (Note that \(0.01\% = 0.0001\,.\)) If you deposit $420 into an account with this interest rate, how long before the account balance is $500?

  5. At a standard roulette table in the US, if you bet on a single number, and the wheel lands on that number, you win 35 times your original bet.

    Suppose you start with $100, and bet it on a single number, and win! You decide to let-it-ride, betting all your winnings on a single number again, and you win again! If you continue doing this, how many times will you have to win before you become the richest person on earth? (Look up how much money this entails.)

Puzzle

There are five houses of different colors next to each other on the same road. In each house lives a woman of a different nationality. Each woman has her favorite drink, her favorite brand of cigarettes, and keeps a particular kind of pet.

The Englishwoman lives in the red house.
The Swede keeps dogs.
The Dane drinks tea.
The green house is just to the left of the white one.
The owner of the green house drinks coffee.
The Pall Mall smoker keeps birds.
The owner of the yellow house smokes Dunhills.
The woman in the center house drinks milk.
The Norwegian lives in the first house.
The Blend smoker has a neighbor who keeps cats.
The woman who smokes Blue Masters drinks bier.
The woman who keeps horses lives next to the Dunhill smoker.
The German smokes Prince.
The Norwegian lives next to the blue house.
The Blend smoker has a neighbor who drinks water.

Who keeps fish?