Question

Once Alice discovers that log10 162 means "the power to which I should raise 10 to get 162," she grows curious about what the graph of a logarithmic function looks like. She realizes there is a different logarithmic function for each base. For example, one such function is defined by the equation y = log2 x.

Alice’s investigations rely heavily on the fact that the equation c = loga b means the same thing as the equation ac = b. This relationship allows her to work with exponential equations, with which she is more comfortable.
For each equation, choose values for x for which you can easily compute the value of y. Plot the resulting points. Choose enough points that you can sketch the graph accurately.
A. y = log2 x
B. y = log3 x
Graph the equation y = log10 x on your calculator.
Compare the graph of the logarithmic function using base 2 with the graphs of logarithmic functions using other bases. In general, how does the graph change as the base increases? Why?
How does the graph of a logarithmic function compare with the graph of the corresponding exponential function?"

Answer 1

The graph of a logarithmic function can be plotted by choosing x values that are powers of the base, resulting in integer y values. As the base increases, the graph of the logarithmic function becomes less steep. The logarithmic graph is the inverse of its corresponding exponential function graph.

Step-by-step explanation:

The relationship c = loga b corresponds to the equation ac = b, which is fundamental in understanding how to graph logarithmic functions like y = log2 x and y = log3 x. To graph these, we select values of x that produce integer results of y. For instance, for y = log2 x, choosing values like x=1, 2, 4, 8, etc., results in y values of 0, 1, 2, 3, etc., because 2 raised to the power of 0, 1, 2, and 3 equals 1, 2, 4, and 8 correspondingly. Similarly, for y = log3 x, using x=1, 3, 9, 27, etc., provides y values of 0, 1, 2, 3, and so on.

When comparing graphs of logarithmic functions with different bases, as the base increases, the growth of the function on the graph becomes less steep. This is because a larger base requires a smaller exponent to reach the same value. For example, log10 x grows slower than log2 x because 10 raised to any power will be larger than 2 raised to the same power.

The graph of a logarithmic function is the inverse of its corresponding exponential function graph. For example, the graph of y = log2 x is the mirror image of the graph of y = 2x across the line y=x. Logarithmic graphs generally increase slowly at first and then more rapidly, whereas exponential graphs start with rapid growth that slows over time.