Download Exponents and logarithms

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 lnx
follows 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 = a
e1 = e
a0 = 1
e0 = 1
a-n
=
1
an
e-2
=
1
e2
EXPONENTS
Using your calculator:
10x: base 10
ex: base e
yx: 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
a n
b
an
= n
b
e 2
3
Try:
=
e 3
2
2a 3
e
e2
32
=
e2
9
LAWS OF EXPONENTS
am
an
m−n
=a
e7
e8
Try:
≡
3e2
e2
an−m
7−8
=e
e4
e2
1
=e
−1
=
1
e
LAWS OF EXPONENTS
Try these:
x−3
y4
a3 b−2
a−5 b7
5a3 b
3ab2
3
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 log
LN: this is loge aka the natural log
x < 1, lnx < 0; x > 1, lnx > 0
Try:
ln 0 =
ln 0.000001 =
ln 1 =
ln 10 =
LAWS OF LOGARITHMS
logb(xz) = logbx + logbz
ln(1*2) = ln1 + ln2 = 0 + ln2 = ln2
ln(3*2) = ln3 + ln2
ln(3*3) = ln3 + ln3 = 2(ln3)
Try:
ln(3*5) =
ln(2x) =
LAWS OF LOGARITHM
x
logb
z
= logbx – logbz
2
loge
3
3
loge
5
= ln2 – ln3
= ln3 – ln5
Try:
ln
ln
2
7
x
7
=
=
LAWS OF LOGARITHMS
logb(xr) = rlogbx for every real number r
loge(23) = 3ln2
loge(32) = 2ln3
Try:
loge42 =
logex3 =
ln3x =
ln and e
Recall, ln is the inverse of e
x
1
1.5
lnx
0
0.40546
elnx
1
e0.40546 = 1.5
3
1.09860
e1.09860 = 3
Try:
x=2
x = 0.009
EXAMPLES OF LOGARITHMS
Try:
w/o calculator lne5
rewrite in condensed form:
2lnx + lny +ln8
3ln5 – ln19
expand:
ln10x3
RADICALS
n
1
n
a a
n
a is called a radical
a is the radicand
n is the index of the radical
is the radical sign
a  2 a by convention and is called square root
LAWS OF RADICALS
Laws of radicals follow the laws of exponents:
n
1
*n


n
n
e   e   e  e
 
1
n
n
Try:
3
8e
4
16
e
mn
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