exponents and logarithms
DESCRIPTION
EXPONENTS AND LOGARITHMS. e. e is a mathematical constant ≈ 2.71828… Commonly used as a base in exponential and logarithmic functions: exponential function – e x natural logrithm – log e x or lnx follows all the rules for exponents and logs. EXPONENTS. - PowerPoint PPT PresentationTRANSCRIPT
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EXPONENTS AND LOGARITHMS
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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
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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 =
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EXPONENTS
Using your calculator:10x: base 10ex: base eyx: base y
Try:e2 =e1.5 =
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LAWS OF EXPONENTS
The following laws of exponents work for ANY exponential function with the same base
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LAWS OF EXPONENTS
aman = am+n
e3e4 = e3+4 = e7
exe4 = ex+4
a2e4 = a2e4
Try: e7e11
eyex
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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
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LAWS OF EXPONENTS
(ab)n = anbn
(2e)3 = 23e3 = 8e3
(ae)2 = a2e2
Try: (ex)5
2(3e)3
(7a)2
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LAWS OF EXPONENTS
Try:
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LAWS OF EXPONENTS
Try:
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LAWS OF EXPONENTS
Try these:
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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)
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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)
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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 =
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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) =
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LAWS OF LOGARITHM
logb= logbx – logbzloge = ln2 – ln3loge = ln3 – ln5
Try:ln = ln =
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LAWS OF LOGARITHMS
logb(xr) = rlogbx for every real number rloge(23) = 3ln2loge(32) = 2ln3
Try:loge42 =logex3 =ln3x =
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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
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EXAMPLES OF LOGARITHMS
Try:w/o calculator lne5
rewrite in condensed form: 2lnx + lny +ln83ln5 – ln19
expand:ln10x3
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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
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LAWS OF RADICALS
Laws of radicals follow the laws of exponents:
Try:
eeeen*
n1n
n1
nn
3 e8
4e16
m n e
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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
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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