Download Test Review Logarithms Exponentials Rational Exponents

WECHS 10th Grade Math
December 16, 2010
Test December 17

Logarithms
◦
◦
◦
◦
◦

Convert between log and exponential form
Evaluate logs – get numerical answer
Expand logs using log properties
Simplify logs using log properties
Solve exponential equation using log
Rational exponents
◦ Convert between radical and exponential form
◦ Evaluate – get numerical answer

Exponential review – calculate growth or decay
Swap between forms:
LOG
EXPONENTIAL
log b a  x
b a
log a  x
10  a
ln a  x
e a
x
x
x
Product Property
 Logb(x·y) = Logbx + Logby
Quotient Property
 Logb(x ÷ y) = Logbx - Logby
Power Property
 Logbxy = y·Logbx
Log of Base
 Logbb = 1
Log of 1
 Logb1=0
When you have the question “logba = x” (b and a
will be numbers), ask “b to what power equals
a?”
Ex:
 Log327 = ?
3 to what power equals 27?
 Log525 = ?
5 to what power equals 25?
 Log .01 = ?
10 to what power equals .01?
If you have a fractional base, flip the fraction over
and make the answer negative.
Log⅓9 = ? Log39 = -x
-x = 2
x = -2

Each factor should be split off into its own
log, using the Product or Quotient Property.
◦ Log6(2xy) = Log62 + Log6x + Log6y

Split into separate logs before using the
Power Property to remove exponents.
◦ Log54x2 = Log54 + Log5x2 = Log54 + 2Log5x

Except – if a power applies to more than one
factor, do that first.
◦ Log(2x)3 = 3Log2x = 3(Log2 + Logx)

The opposite of expanding – combine
everything into a single log if possible.
◦ Logx + Logy + Log6 = Log6xy
x
Logx  Logy  Logz  Log
yz

Move factors outside the log up to be
exponents before combining terms.
◦ 3Logx + 2logy = Logx3 + Logy2 = Logx3y2


If you have an equation with variable in the
exponent, take log of both sides.
Ex: solve 3x = 18
x
log 3  log 18
x log 3  log 18
log 18
x
log 3

A rational exponent is a fraction in the
exponent:
◦ Ex:

x ⅓, y ⅗, z
⅝
This can be written in radical or exponent
form:
m
n
a  a  ( a)
n
m
n
m

Write
27
2
3
in radical form:
2
3
27  ( 27 )

Write
4
( 16 )
3
3
in exponent form:
( 16 )  16
4
2
3
3
4

Take root first if possible so you work with
smaller numbers; then raise to a power.
2
3
27  ( 27 )
3

3
2
2
3
27  3, so, 27  (3)  9
2
You can raise to a power first if it is not
possible to take the root.
3
2
8  8  512  16 2
2
3

xm·xn = x(m+n)
m
x
mn
.
x
n
x

(xm)n = xmn
1
m
.

x

xm

Exponential functions show things that grow
or decay/depreciate at a constant rate:
x
y  A(1  r )


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A is the initial amount (principle).
r is the growth rate (as a decimal)
If it is a growth problem, r is positive and the
number in ( ) is greater than 1
If it is a decay problem, r is negative, and the
number in ( ) is less than 1.