An exponential function is defined by \(k(x)= 15 \boldcdot 2^x\).

1. Show that when \(x\) increases from 1 to 1.25 and when it increases from 2.75 to 3, the value of \(k\) grows by the same factor.
2. Show that when \(x\) increases from \(t\) to \(t+0.25\), \(k(t)\) also grows by this same factor.

How many times does \$1 need to double in value to become \$1,000,000? Explain how you know.

What values could replace the “?” in these equations to make them true?

1. \(\log_{10} 10,\!000 = {?}\)
2. \(\log_{10} 10,\!000,\!000 = {?}\)
3. \(\log_{10} {?} = 5\)
4. \(\log_{10} {?} = 1\)

1. What value of \(t\) would make the equation \(2^t = 6\) true? Write the value exactly.
2. Between what two whole numbers is the solution for \(t\)? Explain how you know.

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

1. \(3^4 = 81\)
2. \(10^0 = 1\)
3. \(4^\frac12= 2\)
4. \(2^t = 5\)
5. \(m^n = C\)