Questions: Which of the following is the graph of an exponential function of the form y=ab^x, where the horizontal axis is a standard scale, and the vertical axis is a logarithmic scale?

Solution Steps

We are looking for the graph of an exponential function \(y = ab^x\) with a standard horizontal scale and a logarithmic vertical scale. This means if we take the logarithm of both sides of the equation, we should get a linear relationship.

Taking the logarithm base 10 of both sides, we have \(\log(y) = \log(ab^x)\) \(\log(y) = \log(a) + x\log(b)\)

Let \(Y = \log(y)\), \(A = \log(a)\), and \(B = \log(b)\). Then the equation becomes: \(Y = A + Bx\) This is a linear equation with slope \(B\) and y-intercept \(A\). Since \(b > 1\), we must have \(B > 0\), so we expect the graph to have a positive slope.

Option C is a straight line with a positive slope, and the vertical axis is a logarithmic scale (1, 10, 100, 1000). Therefore, this option corresponds to an exponential function on a semilog plot.

Option A shows exponential growth, but both axes are linear. Option B also shows exponential growth, but both axes are logarithmic. Option D shows exponential growth, with a linear horizontal axis and what appears to be a logarithmic vertical axis, but it is marked in unusual intervals such as 10, 100, 1000, 10,000, 100,000, 1,000,000, and 10,000,000. This is simply a logarithmic scale, where the intervals increase as powers of 10.

Final Answer