Here is the graph of a logarithmic function of the form \(y = \log_a (x)\).

What value of \(a\) would produce this graph? Explain how you know.

Match each equation with a graph that represents it.

The cost of a medical treatment was \$150 in 1978. The cost increased exponentially by a factor of 1.35 from 1978 to 1983 so that in 1983 the cost was \$202.50. If the cost continues to grow exponentially, in what year could the cost be represented by the equation \(150 \boldcdot (1.35)^3\)?

The equation \(A(w) = 180 \boldcdot e^{(0.01w)}\) represents the area, in square centimeters, of a wall covered by mold as a function of \(w\), time in weeks since the area was measured.

Explain or show that we can approximate the area covered by mold in 8 weeks by multiplying \(A(7)\) by 1.01.

Solve each equation without using a calculator. Some solutions will need to be expressed using log notation.

1. \(10^{(n-3)} = 10\)
2. \(\frac12 \boldcdot 10^x = 0.05\)
3. \(10^{\frac13 t} = 100\)
4. \(10^{2x} = 48\)

Technology required. The population of Mali can be modeled by \(m(t)= 17 \boldcdot e^{(0.03t)}\). The population of Saudi Arabia can be modeled by \(s(t) = 31 \boldcdot e^{(0.015t)}\). In both models, \(t\) represents years since 2014, and the populations are measured in millions.

1. Which country model shows a higher population in 2014? Explain how you know.
2. Which country's model has a higher growth rate? Explain how you know.
3. Use graphing technology to graph both equations on the same axes.
4. Do the two graphs intersect? If so, estimate their point of intersection and explain what it means in this situation. If not, explain what it means that the two graphs don’t intersect.