(a) Explain why we know ex has an inverse function—let’s call it log x—defined on the strictly positive real numbers and satisfying(i) log(ey) = y for all y ∈ R and(ii) elog x = x, for all x > 0.(b) Prove (log x) = 1/x. (See Exercise 5.2.12.)(c) Fix y > 0 and differentiate log(xy) with respect to x. Conclude that log(xy) = logx + log y for all x, y > 0. (d) For t > 0 and n ∈ N, tn has the usual interpretation as t · t · · · t (n times). Show that(2) tn = en log t for all n ∈ N. Part (d) of the previous exercise is the pivotal formula because the expression on the right of the equal sign is meaningful if we replace n with x ∈ R. This is our cue to use the identity in (2) as a template for the definition of tx on all of R.
Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
(a) Explain why we know ex has an inverse function—let’s call it log x—defined on the strictly positive real numbers and satisfying
(i) log(ey) = y for all y ∈ R and
(ii) elog x = x, for all x > 0.
(b) Prove (log x) = 1/x. (See Exercise 5.2.12.)
(c) Fix y > 0 and differentiate log(xy) with respect to x. Conclude that log(xy) = logx + log y for all x, y > 0. (d) For t > 0 and n ∈ N, tn has the usual interpretation as t · t · · · t (n times). Show that
(2) tn = en log t for all n ∈ N. Part (d) of the previous exercise is the pivotal formula because the expression on the right of the equal sign is meaningful if we replace n with x ∈ R. This is our cue to use the identity in (2) as a template for the definition of tx on all of R.
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