logs equal the the inverse of an exponential function is a logarithmic function. logarithmic...
TRANSCRIPT
The inverse of an exponential function is a logarithmic function.
Logarithmic Function
x = log a y
read: “x equals log base a of y”
y = bx x = logby
These two equations are equivalent
We can convert exponential equations to logarithmic equations and vice versa, using
this:
Now that we can convert between the two forms we can simplify logarithmic expressions. Without a Calculator!
Simplify
1) log2 32
2) log3 27
3) log4 2
4) log3 1
2? = 32
3? = 27
4? = 2
3? = 1
? = 5
? = 3
? = 0.5
? = 0
“What is the exponent of that gives you 32?”
“What is the exponent of 3 that gives you 27?”
We can also use these two forms to help us solve for an inverse.The steps for finding an inverse are the
same as before.Easy as 1, 2, 3…1-Rewrite2-Switch x and y3-Solve for y
Example: Find the inverse
1. rewrite (no-need) 8xy 2. Switch and 8yx y x 3. Solve for y 8yx
8xy
Rewrite in log form : 8yx 8 log x y
8 log is the inverse of 8 xy x y
An inverse you just have to know
Ln and are inverses
They undo each other
1.
2.
xe
ln( )xe x
ln( )xe x
Example: Find the inverse
( ) ln( 2)f x x
1. Rewrite: ln( 2)y x
2. Switch and : ln( 2)x y x y
3. Solve for : ln( 2)y x y
ln( 2)
Inverse of ln is
So exponentiate both sides base e: e
x
x y
x e
e
e 2x y
e 2xy
1 e 2xf x
Essential Question: How do I graph & solve exponential and
logarithmic functions?
Daily Question: How do you expand and condense
logs?
Change-of-Base Formula
loglog
logab
b
xx
a
Let a, b, and x be positive real numbers such that a1 and b1.
loglog
loga xx
a 10
10
logln
lna xx
a
Ex. 1
a) Evaluate using the change-of-base formula. Round to four decimal places.
log3 7 log
log10
10
7
3
Ex. 2You can do the same problem using natural logarithms.
a) Evaluate using the natural logarithms. Round to four decimal places.
log3 7 ln
ln
7
3
Properties of Logarithms
log ( ) log loga a auv u v
ln( ) ln lnuv u v
log log loga a a
u
vu v
ln ln lnu
vu v
ln lnu n un
log logan
au n u
Product Property
Quotient Property
Power Property
Ex. 5Condense.
1
23 110 10log log ( )x x log log ( )10
12
1031x x
log log ( )10 1031x x
log [ ( ) ]1031x x
2 2ln( ) lnx x ln( ) lnx x2 2
ln( )x
x
2 2
a)
b)
Ex. 6Use and to evaluate the logarithm.
3log 5 1.465 3log 6 1.631
3
6A. log
5
3 3 log 6 log 5
1.631 1.465
0.166
3B. log 30
3 log 5 6
3 3 log 5 log 6
1.465 1.631
3.096
3C. log 36
23 log 6
3 2 log 6
2 1.631
3.262
Ex. 7 Use the properties of logarithms to verify that -ln ½ = ln 2
-ln ½ =
-ln (2-1) =
-(-1) ln (2) =
ln (2) = ln 2