College Algebra 9th txtbk

SECTION 5–3 Logarithmic Functions 359
SOLUTIONS (A) log e 1 0
Property 1 (B) log 10 10 1
Property 2
(C) log Property 3 (D) log 10 0.01 log 10 10 2 e e 2x1 2x 1
2 Property 3
(E) Property 4 (F) e log e x
10 log 10 7 7
2 x 2
Property 4
MATCHED PROBLEM 5 Simplify, using the properties in Theorem 2.
(A) log 10 10 5 (B) log 5 25 (C) log 10 1
(D) (E) 10 log 10 4
log e e mn (F)
e log e (x 4 1)
Z Common and Natural Logarithms
To work with logarithms effectively, we will need to be able to calculate (or at least approximate)
the logarithms of any positive number to a variety of bases. Historically, tables were
used for this purpose, but now calculators are used because they are faster and can find far
more values than any table can possibly include.
Of all possible bases, there are two that are used most often. Common logarithms are
logarithms with base 10. Natural logarithms are logarithms with base e. Most calculators
have a function key labeled “log” and a function key labeled “ln.” The former represents
the common logarithmic function and the latter the natural logarithmic function. In fact,
“log” and “ln” are both used in most math books, and whenever you see either used in this
book without a base indicated, they should be interpreted as follows:
Z LOGARITHMIC FUNCTIONS
y log x log 10 x
y ln x log e x
Common logarithmic function
Natural logarithmic function
ZZZ EXPLORE-DISCUSS 2
(A) Sketch the graph of y 10 x , y log x, and y x in the same coordinate
system and state the domain and range of the common logarithmic function.
(B) Sketch the graph of y e x , y ln x, and y x in the same coordinate
system and state the domain and range of the natural logarithmic function.
EXAMPLE 6 Calculator Evaluation of Logarithms
Use a calculator to evaluate each to six decimal places.
(A) log 3,184 (B) ln 0.000 349 (C) log (3.24)
SOLUTIONS (A) log 3,184 3.502 973 (B) ln 0.000 349 7.960 439
(C) log (3.24) Error
Why is an error indicated in part C? Because 3.24 is not in the domain of the log function.
[Note: Calculators display error messages in various ways. Some calculators use a more
advanced definition of logarithmic functions that involves complex numbers. They will