EXPONENTS AND LOGARITHMS

e

e is a mathematical constant≈ 2.71828…

Commonly used as a base in exponential and logarithmic functions:

exponential function – ex

natural logrithm – logex or lnxfollows all the rules for exponents and logs

EXPONENTS

an where a is the base and x is the exponent

an = a · a · a · … · a e3 = e * e * e

a1 = ae1 = e

a0 = 1e0 = 1

a-n = e-2 =

EXPONENTS

Using your calculator:10x: base 10ex: base eyx: base y

Try:e2 =e1.5 =

LAWS OF EXPONENTS

The following laws of exponents work for ANY exponential function with the same base

LAWS OF EXPONENTS

aman = am+n

e3e4 = e3+4 = e7

exe4 = ex+4

a2e4 = a2e4

Try: e7e11

eyex

LAWS OF EXPONENTS

(am)n = amn

(e4)2 = e4*2 = e8

(e3)3 = e3*3 = e9

(108)5 = 108*5 = 1040

Try: (e2)2

(104)2

LAWS OF EXPONENTS

(ab)n = anbn

(2e)3 = 23e3 = 8e3

(ae)2 = a2e2

Try: (ex)5

2(3e)3

(7a)2

LAWS OF EXPONENTS

Try:

LAWS OF EXPONENTS

Try:

LAWS OF EXPONENTS

Try these:

LOGARITHMS

The logarithm function is the inverse of the exponential function. Or, to say it differently, the logarithm is another way to write an exponent.

Y = logbx if and only if by = x

So, the logarithm of a given number (x) is the number (y) the base (b) must be raised by to produce that given number (x)

LOGARITHMS

Logarithms are undefined for negative numbers

Recall, y = logbx if and only if by = x

blogbx = x eloge2 = eln2 = 2 (definition )

logaa = 1 logee = lne = 1 (lne = 1 iff e1 = e)

loga1 = 0 loge1 = ln1 = 0 (ln1 = 0 iff e0 = 1)

LOGARITHM

Using your calculator:LOG: this is log10 aka the common logLN: this is loge aka the natural logx < 1, lnx < 0; x > 1, lnx > 0

Try:ln 0 =ln 0.000001 = ln 1 = ln 10 =

LAWS OF LOGARITHMS

logb(xz) = logbx + logbzln(1*2) = ln1 + ln2 = 0 + ln2 = ln2ln(3*2) = ln3 + ln2ln(3*3) = ln3 + ln3 = 2(ln3)

Try:ln(3*5) = ln(2x) =

LAWS OF LOGARITHM

logb= logbx – logbzloge = ln2 – ln3loge = ln3 – ln5

Try:ln = ln =

LAWS OF LOGARITHMS

logb(xr) = rlogbx for every real number rloge(23) = 3ln2loge(32) = 2ln3

Try:loge42 =logex3 =ln3x =

ln and e

Recall, ln is the inverse of e

Try:x = 2x = 0.009

x lnx elnx

1 0 11.5 0.40546 e0.40546 = 1.53 1.09860 e1.09860 = 3

EXAMPLES OF LOGARITHMS

Try:w/o calculator lne5

rewrite in condensed form: 2lnx + lny +ln83ln5 – ln19

expand:ln10x3

RADICALS

n1

n aa n a is called a radical

a is the radicand n is the index of the radical

is the radical sign

by convention and is called square root2 aa

LAWS OF RADICALS

Laws of radicals follow the laws of exponents:

Try:

eeeen*

n1n

n1

nn

3 e8

4e16

m n e

SCIENTIFIC NOTATION

Numbers written in the form a x 10b

when b is positive – move decimal point b places for the right

when b is negative – move decimal point b places to the left

Reverse the procedure for number written in decimal form

Follows the laws of exponents

EXAMPLES OF SCIENTIFIC NOTATION

1,003,953.79 1.00395379 x 106

-29,000.00 -2.9 x 104

0.0000897 8.97 x 10-5