Here arc a few more values of Logz:

Log10 = I s i n c e 10™ = 10,

Log 1tltt.w˜ = 5 SinCc 1 0 ™ = 100,000,

Log I = 0 since lo” = I,

L o g 625 = 2 . 7 9 5 8 8 since 10™˜˜45K”” = 625.

For most values of z, you™ll have to use a calculator or table of logarithms to

evaluate Logz.

One forms the graph of the invcrsc function S-™ by reversing the roles of the

horizontal and vertical axes in the graph of f. In other words, the graph of the

invcrsc of a function y = l(x) is the reflection of the graph off across the diagonal

{x = y}, because (y, z) is a point on the graph off-˜™ if and only if (z, y) is a point

on the graph off. In Figure 5.4, WC hew drawn the graph of y = lt?™ and reflected

it across the diagonal {.r = v} to draw the graph of y = Logx.

Since the negative “.r-axis“ is a horizontal asymptote for the graph ofy = IO™,

the ncgatiw “y-axis” is a vertical asymptote for the graph ofy = Log-r. Since 10˜™

grows wry quickly, Logv grows very slowly. At I = 1000. Logx is just at y = 3;

at x cquals a million. Logx has just climbed toy = 6. Finally, since for Avery x,

IO™ is a positive number. Logi is only dcfmcd for x Z> 0. Its domain is R--. the

set of strictly positive numhcrs.

90 E X P O N E N T S A N D L O G A R I T H M S [51

Base e logarithms

Since the exponential function exp(n) = e™ has all the properties that lw has, it

also has an inverse. Its inverse works the same way that Logx does. Mirroring

the fundamental role that e plays in applications, the inverse of e™ is called the

natural logarithm function and is written as Inx. Formally,

Inx=y - e)=x;

lnx is the power to which one nust raise e to get x. As we saw in general in (3),

this definition can also be summarized by the equations

e™“™ = x a n d I”$ =x. (4)

The graph of ti and its reflection across the diagonal, the graph of In x, are similar

to the graphs of 1V and Logx in Figure 5.4.

Example 5.2 Let™s work o”t some examples. The natural log of 10 is the power

of r that gives 10. Since r is a little less than 3 and 32 = 9, e2 will be a bit less

than 9. We have to raise e to a power bigger than 2 to obtain 10. Since 3™ = 27,

e3 will be a little less than 27. Thus, we would expect that In 10 to lie between

2 and 3 and somewhat closer to 2. Using a calculator, we find that the answer

to four decimal places is I” 10 = 2.3026.

We list a few more examples. Cover the right-hand side of this table and try

to estimate these natural logarithms.

Ine =l since e™ = E;

I” 1 =o since e0 = 1;

si”ce e-2.3o25.

InO. = -2.3025... = 0.1;

si”ce pw

In 40 = 3.688. = 40;

si”cc pa...

I”? = 0.6931 ” = 2.

EXERCISES

5.3 First rstimate the Iollowing logarithms without a calculator. Then, use your calculator

to compute an answer correct to four decimal places:

a) Log snn. c) Log 1234,

h) Log5, d) Loge,

f) I” 100,

e) In30. g) In.3. h) In T.

L5.41 PROPERTIES OF EXPAND LOG 91

5.4 Give the exact values of the following logarithms without using a calculator:

c) Log(billion),

u) Log IO, h) Logo.onl,

4 lo&& e) log, 36 f) b, O.&

i) In 1.

h) hh>

8 W™L

5.4 PROPERTIES OF EXP AND LOG

Exponential functions have the following five basic properties:

(1) ar .a˜ = a™+˜,

(2) a -r = I / a ™ ,

( 3 ) d/d = a ” ,

(4) (a™)˜ = a”, and

(Pi) a” = 1.

Propertics 1, 3, and 4 are straightforward when I and s are positive integers. The

definitions that a-” = I/a”, a” = 1, a I”™ is the nth root of a, and a”“” = (a”“)”

are all specifically designed so that the above five rules would hold for all real

numbers I and S.

These tive properties of exponential functions are mirrored hy five correspond-

ing properties of the logarithmic functions:

(I) log(r ˜5) = logr + logs,

(2) log( I /s) = - logs,

(3) log(r/s) = logr - logs.

(4) log? = sl”gr,and

(5) log I = 0.

The fifth property “f logs follows directly from the fifth property of OI and the

fact that a™ and log,, are inverses of each other. To prove the other four properties,

let LI = log,, r and v = log,, S, so that r = a” and s = a™. Then, using the fact that

l”g,<(n˜˜) = x. WC find:

(1) log(r .s) = log(u” a”) = log(a”˜“˜) = u + I™ = log I + log ˜,

( 2 ) log(l/.s) = log(l/a™) = log(C) = -I™ = ˜ l o g s .

( 3 ) log(r/s) = log(a”/a™) = log(a”˜˜˜) = u v = logr l o g s ,

= s.logr.

(4) log r( = b&l”) = log a” = us

Logarithms are especially useful in bringing a variable n that occurs as an

exponent hack down t” the base line whcrc it can he m”˜e easily manipulated.

151

EXPONENTS AND LOGARITHMS

92

Example 5.3 To solve the equation 2 5x = 10 for x, we take the Log of both sides:

Log2™X=LoglO OT 5x.Log2=1.

i

It follows that

1

x = __ z.6644.

5 Log2

We could have used 1” instead of Log in this calculation

Example 5.4 Suppose we want to find o”t how long it takes A dollars deposited

in a saving account to double when the annual interest rate is r compounded

continuously. We want to solve the equation

2A = Ae” (5)

for the unknown f. We first divide both sides of (5) by A. This eliminates A

from the calculation-a fact consistent with o”r intuition that the doubling

time should be independent of the amount of money under consideration. To