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Math 32 - Section 101 and 102 - March 18, 2015 GSI: Benjamin Filippenko Quiz #7 Name: Solutions Box your answers. Show your work. You have 20 minutes. 1. Order the following numbers from smallest to largest. Your answer should be a list of all of these numbers with ”<” and ”=” symbols inbetween them. a) 3 √ b) 2 log3 2 c) log10 (105 ) d) (log2 6) · (log6 8) of base forSolution: First of all, log10 (105 ) = 5, by the definition of log. Next, √ by the change √ mula, we have (log2 6)(log6 8) = log2 8 = 3. Finally, notice 2 log3 2 = log3 ( 2)2 = log3 2 < 1, since log3 3 = 1. So we have (b) < (a) = (d) < (c). 2. Find a function f (x) with exponential growth such that the function g(x) = log5 f (x) is a line with slope equal to 2 and y-intercept equal to 1. Solution: We know that f (x) = cbkx for some numbers c, b, k, because this is the definition of having exponential growth. So then g(x) = log5 f (x) = log5 (cbkx ) = log5 c + log5 (bkx ) = x(k log5 b) + log5 c. This is a line with slope k log5 b and y-intercept log5 c. We want the y-intercept to be 1, so solving log5 c = 1 yields c = 5. We want the slope to be 2, so we need k log5 b = 2. One possible solution of this is k = 2, b = 5. Thus f (x) = 5 · 52x . Note that there are other choices of c, b, k that work, and these correspond to equivalent ways of writing the function f (x). For example, f (x) = 5 · (25)x corresponds to c = 5, b = 25, k = 1. 3. Extra credit: why is it true that log ab = log a + log b ? Solution: All ways of explaining this come down to the fact that exponents add when we multiply: xa xb = xa+b . The fact that log ab = log a + log b is just the corresponding rule for logs. Here is one possible explanantion: By definition of log, we have 10log ab = ab and 10log a = a and 10log b = b. So, 10log a+log b = 10log a 10log b = ab = 10log ab , and then comparing the right and left sides of the above equation yields log a + log b = log ab, as desired.