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Unformatted text preview: Sec. 4.14.3: ExponenTiel FuncTions An expenenTiel funeTien is one in which The variable is in The expenenT and The base is n censTunT. ETC»): bx whee! b>0. b7“ éjj: 7:L NeTe:
 The base musT be u pesiTive number‘r exeepT 1.
 The sTundurd expenenTiel funeTien is ene—Te—ene.  The neTuml eernenTiel funeTign hes base e (e = 2.7182 8182 8....), en ianiennl number.
5 a1 Deriwd'iw 5 '* The logarithmic function to the base a, where a > 0 and a :t 1
is defined: y = logax ifand only ifx = ay
_ logarithmic form exponential form
When you convert an exponential to log form, notice that the exponent in the exponential becomes what the log is equal to. I—l
Convert to log form: : 42 10g416 = 2 Convert to exponential form: 1
10g2 g : —3 :16 EE'E'EYFBNENFEE With this in mind, we can answer questions about the log: This is asking for an exponent. What
10g2 = 4 exponent do you put on the base of 2 to
get 16? (2 to the what is 16?) What exponent do you put on the base of
_2 3 to get “9? (hint: think negative) 0 What exponent do you put on the base of
4 to get 1? 1 When working with logs, rewrite any 10g3 32 = E radicals as rational exponents. What exponent do you put on the base of
,x / 3 to get 3 to the 1I2? (hint: think rational) Remember our natural base “e”?
We can use that base on a log. log 2 1 What exponent do you put I 3 on e to get 2.7182828? 111
Since the log with this base occurs In : 1 in nature frequently, it is called the natural log and is abbreviated In. Your calculator knows how to find natural logs. Locate
the In button on your calculator. Notice that it is the
same key that has ex above it. The calculator lists
functions and inverses using the same key but one of
them needing the 2nd (or inv) button. 8” / Another commonly used base is base 10.
A log to this base is called a common log. Since it is common, if we don't write in the base on a log
it is understood to be base 10. _ What exponent do you put
10g10?— 2 on10to get100? 10g _ _ 3 What exponent do you put 1000 _ on 10 to get 111000? This common log is used for things like the richter
scale for earthquakes and decibles for sound. Your calculator knows how to find common logs. Locate the log button on your calculator. Notice that it is the same key that has 10" above it. Again, the calculator lists functions and inverses using the same
I key but one of them needing the 2nd (or inv) button. Properties
of Logarithms Since logs and exponentials of the same base are
inverse functions of each other they “undo” each other. ax f_1(x)=10ga x Rememberthat: fof_1 :xand f_1 of = x This means that: fOf—l _ 1‘3ng Z x inverses
“undo” each
each other P315=5 Properties of Logarithms CONDENSED = EXPANDED (these properties are based on rules of
exponents since logs = exponents) Using the log properties, write the expression as a
sum andlor difference of logs (expand). b4
10g 6 a 21036 3 2
C 6154
3
C3
When working with logs, rewrite anWals as rational
exponents. 2 using the second property: 10g 6 6:54 — 10g 6 C 3 M
10 —:lo M—lo N \
go N go: go 2
using the first property: 10g 6 a +10g754 —10g‘673 logﬂI W : 1s:ngI M + logcI N
2 using the third property: 10g6 a + 410g6 b — —10g6 C
logaMr :rlogaM 3 Using the log properties, write the expression as a
single logarithm (condense). 210%.: 3C—i10g3 y VJ using the third property: 10g3 x2 _10g3 yE logﬂMr : rlogﬂM
this direction
«1—
2
1: using the second property: 10g 3 —1 M
_ : _ 2
logﬂ loga M logﬂ N y this direction
.1— More Properties of Logarithms This one says if you have an equation, you can take
the log of both sides and the equality still holds. IfM = N, then logaM =10ga N If logaM =10ga N,thenM =N This one says if you have an equation and each side
has a log of the same base, you know the "stuf " you are taking the logs of are equal. 10g28= 3 {2 to the what is 8’?) 10g216 = 4 (2 to the what is 16’?) 10g210 #332 (2 to the what is 10’?) Check by
putting 2332 in
your calculator (we rounded so
it won't be exacD There is an answer to this and it must be more than 3 but less than 4, but
we can't do this one in our head. Let's put it equal to X and we'll solve for X. Change to
exponential form. log210=x
2x=10 use log property & take log of
both sides (we'll use common log) IfM =N,thenlogaM =1ogaN 10g 2‘ =10g10 use 3rd log property
log:erI Mr = Flog”I M solve for X by
dividing by log 2 x10g2=10g10 : loglO % 332
log2 use calculator to x
approximate If we generalize the process we just did
we come up with the: ChangeofBase Formula 10: [email protected] — leg—M — loga _ lnM — Ina The base you change to can “common”
be any base so generally 10g base 10
we’ll want to change to a LOG base so we can use our
calculator. That would be either base 10 or base e. “natural” 10% base a I? Emlm1wmmls TI'BE Use the ChangeofBase Formula and a calculator to approximate the logarithm. Round your answer to three
decimal places. 10g316 Since 32 = 9 and 33 = 27, our answer of what exponent
to put on 3 to get it to equal 16 will be something
between 2 and 3. 1n16 log316= 1113 a 2.524 put in calculator ...
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 Fall '10
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