Applied Calculus Math 215 - University of Hawaii

26 CHAPTER 1. SOME BACKGROUND MATERIAL
Furthermore, log a(u) =log a(v) implies that
u = a log a (u) = a log a (v) = v.
This verifies the remaining claim in the theorem.
Corresponding to the exponential laws in Theorem 1.13 on page 22 we
have the laws of logarithms. Some parts of the theorem are proved in Section
4.11. The other parts are assigned as exercises below.
Theorem 1.19 (Laws of Logarithms). For any positive real number a =
1, for all positive real numbers x and y, and any real number z
loga(1) = 0
loga(a) = 1
loga(xy) = loga(x)+loga(y) loga(x/y) = loga(x)−loga(y) loga(x z ) = zloga(x) Because the exponential and logarithm functions are inverses of each
other, their rules are equivalent. In the following exercises you are asked to
verify this.
Exercise 24. Assume the exponential laws and deduce the laws of logarithms.
Exercise 25. Assume the laws of logarithms and deduce the exponential
laws.
To show you how to solve this kind of problem, we deduce one of the
exponential laws from the laws of logarithms. Observe that
log a(a x a y )=log a(a x )+log a(a y )=x+y=log a(a x+y ).
The first equation follows from the third equation in Theorem 1.19, and the
remaining two equations hold because of the way the logarithm function
is defined. Comparing the outermost expressions, we deduce from Theorem
1.18 the third exponential law:
Exercise 26. Assume that
Show that
a x a y = a x+y .
log a 1=0 and log a(xy) =log a(x)+log a(y).
log a(x/y) =log a(x)−log a(y).