Using the fact that \(2^{10} = 1024\), Tyler estimates that \(2^{20}\) is about 1,000,000, and \(\log_2(1,\!000,\!000)\) is about 20. Do you agree with Tyler? Show or explain your reasoning.

For each logarithmic equation, write an equivalent equation in exponential form.

1. \(\ln 618 = p\)
2. \(\ln q = 2\)
3. \(\ln 100 = t\)
4. \(\ln (e^3) = 3\)

Technology required. The function \(f\) given by \(f(t) = 10e^{0.07t}\) models the balance in a bank account, in thousands of dollars, \(t\) years after it was opened.

1. What was the opening balance?
2. About when does the account balance reach 1,000,000 dollars? Explain or show how you know.

The function \(f\) is given by \(f(x) = 20 \boldcdot e^x\).

1. Write an equation of an exponential function \(g\) whose graph meets the graph of \(f\) for a positive value of \(x\).
2. Write an equation of an exponential function \(h\) whose graph does not meet the graph of \(f\) for any positive value of \(x\).

A bank account had a balance of \$100. Because of the interest accumulated over time, the balance doubles every decade. No withdrawals or other deposits are made.

1. To find out when the account will have a balance of \$1,000, Diego wrote: \(100 \boldcdot 2^t =1,\!000\) and Mai wrote: \(t = \log_2 \left(\frac{1,\!000}{100}\right)\). Show that the equations are equivalent and have the same solution.
2. Use either of the strategies to find out when the account will have the following amounts. Show your reasoning.
   1. \$5,000
   2. \$12,000
3. Write an equation that shows when (at what value of \(t\)) the account will have \(x\) dollars.