1. Section 1.8 Logarithmic functions

2. Logarithmic functions
Definition: Let b > 0 and b not equal to 1. Then y is the logarithm of x to the base be written: y = logbx if and only if by = x

3. Example
24 = x
3x = 9
logb16 = 4
log5 x = 2

4. Example
434 = 81
½361/2 = 6
050 = 1

5. Basic logarithmic properties
Why? b1 = b
Why? b0 = 1
Why? bx = bx
Also, 𝑏 𝑙𝑜𝑔 𝑏 𝑥 =𝑥

6. Other logarithms logb1 = 0 log 1 = 0 ln 1 = 0 logbb = 1 log 10 = 1
Common Logarithm: (base 10)
log x = y  10y = x
Natural Logarithm: (base e)
logex = y ln e = y  ey = x
General Properties
Common Logarithms
Natural Logarithms
logb1 = 0
log 1 = 0
ln 1 = 0
logbb = 1
log 10 = 1
ln e = 1
Logbbx = x
log 10x = x
ln e x =x
𝑏 𝑙𝑜𝑔 𝑏 𝑥 =𝑥
10logx = x
elnx = x

7. log216= 2 log8 + logx + logy logb (xy) = logb x + logb y
Log of a Product:
logb (xy) =
logb x + logb y
Examples:
log216= 2
log4 8 + log4 2 =
log8 + logx + logy
log 8xy =

8. Log5125=3 log 5 + log m – log n logb = logb x - logb y
Log of a Quotient:
logb = ÷ ø ö ç è æ y x
logb x - logb y
Examples:
Log5125=3
log log5 3 =
log = n m 5
log 5 + log m – log n

9. 7(logr + logt) logb xn = n · logb x Log of a Power: Examples:
log (rt)7 =

10. Use the properties to rewrite equations without logs
Consider:
log R = 4log x - log 3
logR = logx4 – log3
logR = logx4/3
R = x4/3

11. log32/log8 = 5/3 log18/log5 = 1.80 Change of base theorem
logba = log (a) /log (b)
log832 =
log518 =
Can also use to graph on calculator:
y = log2xy = log(x)/log(2)
log32/log8 = 5/3
log18/log5 = 1.80

12. Graphing logarithms: y = log2x
To graph y = logb x
Algebraically:
Rewrite as an exponential equation: by = x
Make an x/y table, filling in y first.
Graph points.
Using the calculator
Re-write using change of base theorem
Input y1 = log(x)/log(b) x Y -1 1 2 4

13. Asymptotes (line that graph approaches, but does not touch)
Properties of y = logbx
y = log bx OR
x = by
y = bx
Domain
Range
Asymptotes (line that graph approaches, but does not touch)
Point on all graphs
inverses
x > 0
All reals
All reals
y > 0
x = 0
y = 0
(1, 0)
(0, 1)

14. Example

15. The problems are on your hand-out!
Your Turn…