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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.