2-3 Logarithmic Functions
The common log of a number is that exponent (or power)
to which 10 must be raised to obtain the number.
Notation:
y = log (x) or y = log x
"y = log of x"
Example:
 log (1000) = . . .
 . . . the power to which 10 must be raised to obtain 1000
 so log(1000) = 3
Another way to put it: 3 = log(1000) because 103 = 1000
y = log(x)
means
y
10 = x
log x + 2  log(x + 2)
log x + 2 = log(x) + 2
Be CAREFUL!
Keep this in mind at all times:
A LOG IS AN EXPONENT !!!!
Practice:
x
log x
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1
10
1000
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p. 1
Since logs are defined using exponentials, any “log x” has
an equivalent “exponent” form, and vice-versa.
Going from log form to exponent form (a visual scheme)
Ex.
LOG FORM

log5 13 = x

EXPONENT FORM
5x = 13
Remember – a log is an exponent
___________________________________________
Going from exponent form to log form
EXPONENT FORM
Ex. y14 = x

==>
LOG FORM
log y x = 14
Remember -- a log is an exponent
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p. 2
Inverse properties of logarithms
Think about …
log 10x :
log 10x is the power to which we must raise 10 to get 10x
which is … x! so …
log 10x = x
Think about …
10log x :
log x is the power to which we must raise 10 to get x
so …
10log x = x
Inverse properties of log
log 10x = x
10 log x = x
Notes:
 x can be any expression
 the base of the log (here 10) must match the base of the
exponential (also 10)
Logs using other bases
For any given base b (b > 0), there is a corresponding "log
to the base b" function, logb.
Example:
 y = log2(x) (or log2 x) "log to the base 2 of x"
 y is the exponent to which the base (2) is raised to get x
3
 so log2 8 = 3, because 2 = 8
The common bases for logs are: 10 and e (see calculator)
Conventions
 instead of writing log10 35, you can write log 35
 instead of writing loge 35, you can write ln 35
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p. 3
Inverse properties for other bases
It is easily seen that the inverse properties hold for all
bases:
Inverse properties of loga
loga ax = x
a
log x
a
=x
Remember, x can be any expression (***)
So, for our favorite bases of 10 and e, we have:
(1) log 10x = x
(2) ln ex = x
10log x = x
eln x = x
as before, but also
The bottom line
What would log 3.2x be? Notice that the base of the log
(10) and the base of the exponential (3.2) do not match, so
the inverse property cannot be applied directly. But …
by the inverse property, 10log 3.2 = 3.2 (**), so
log 3.2x = log (10log 3.2)x
= log 10 (log3.2)x
= (log 3.2)x
= x log 3.2
by substituting (**)
by “power of a power rule”
by *** above
In general, we have:
log ax = x log a
(log of a power property)
This is all you need to remember for what follows.
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p. 4
Solving exponential equations
Problem: solve the equation 30(1.06x) = 100 (i.e., find a
value of x that satisfies the equation).
This is an exponential equation, because “x” appears in an
exponent.
How to solve this? You need to get “Mr. x” by himself on
the left-hand-side of the equation to solve it. But “Mr. x”
is incarcerated up there as an exponent, and no amount of
+-÷, or taking of powers or roots will get him out (try it!).
The tool that will work is logarithms. Watch!
(1) 1.06x = 100/30
get the exponential by itself
(2) log 1.06x = log(100/30) take the log of both sides
(3) x log 1.06 = log(100/30) log of a power property
Notice how “x” has now escaped from its prison as an
exponent, leaving us with a run-of-the-mill linear equation
to solve:
(4) x = (log(100/30))/log 1.06  20.662 (via calculator)
Let’s check it on the calculator:
1.0620.662 =?? 3.333
100/30 =?? 3.333 YES! It works!
Note: I did no intermediate arithmetic as I went through
the solution process, rather, saving it all to the end and
evaluating the whole result on the graphing calculator.
This is advisable, because it minimizes the propagation of
round-off error, and your answers will be close to the book
answers.
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