1. 5.3 Logarithmic Functions & Graphs
Find common logarithms and natural logarithms with and without a calculator.
Convert between exponential equations and logarithmic equations.
Change logarithmic bases.
Graph logarithmic functions.
Solve applied problems involving logarithmic functions.
Simplify expressions of the type logaax and

2. A logarithm is just an exponent.
Our first question
What is a logarithm ?
A logarithm is just an exponent.
Evaluate: log39

3. Logarithmic Function, Base b
We define y = logb x as that number y such that x = by, where x > 0 and b is a positive constant other than 1.
We read logb x as “the logarithm, base b, of x.”
A logarithm is an exponent! 3

4. Finding Certain Logarithms - Example
Find each of the following logarithms.
log10 10,000
log
log8 8
log216
log6 1
log2(1/4) 4

5. Basic Properties of Logarithms
For b > 0, and b ≠ 1
Logb1 = 0
Logbb = 1
Logbbx = x
Simplify

6. Basic Properties of Natural Logarithms
Evaluate the following.
ln (1)
ln (e)
ln (ex)
= 0
= 1
= x
= x

7. Logarithms Convert each of the following to a logarithmic equation.
A logarithm is an exponent!
Convert each of the following to a logarithmic equation.
a) 16 = 2x b) 10–3 = c) et = 70 7

8. Example
Rewrite in logarithmic form.
64 = 43
73 = x

9. Logarithms A logarithm is an exponent!
Convert each of the following to an exponential equation.
a) log 2 32= 5 b) log b Q= c) x = log t M 9

10. Example
Rewrite in exponential form
log416 = 2
n=logmx

11. Logarithms
A base 10 logarithm is known as the common logarithm. We denote log10x = log x A base e logarithm is known as the Natural logarithm. We denote logex = ln x

12. Example
Evaluate
log 10000
log(1/1000)
log 512
log(-10)

13. Example
Find each of the following natural logarithms on a calculator. Round to four decimal places.
a) log 645,778
b) ln
c) log (5)
d) ln e
e) ln 1 13

14. The Change-of-Base Formula
For any logarithmic bases a and b, and any positive number M,
Find log5 8 using common logarithms.
Find log5 8 using natural logarithms. 14

15. Logarithmic Functions
Log functions are inverses of exponential functions (y = 2x ).
We can draw the graph of the inverse of an exponential function by interchanging x and y. (x = 2y ). 15

16. Graphs of Logarithmic Functions
, where b > 0 and b ≠ 1.
Domain: (0, ∞)
Range: (-∞, ∞)
x-intercept is (1,0)
No y-intercept
The y-axis is a vertical asymptote (x = 0).
f is 1-1 (one-to-one)
Graph passes through the point (b, 1)
If b > 1, the function is increasing.
If 0 < b < 1, the function is decreasing.

17. Graphs of Logarithmic Functions - Example
Graph: y = f (x) = log5 x.
Solution:
y = log5 x is equivalent to x = 5y. Select y and compute x. 17

18. Example
Describe how each graph can be obtained from the graph of y = ln x. Give the domain and the vertical asymptote of each function.
a) f (x) = ln (x + 3) 18

19. Example f(x) = ln(x-4) + 2 y = -lnx - 2
For each function, state the transformations applied to y = lnx. Determine the vertical asymptote, and the domain and range for each function.
f(x) = ln(x-4) + 2
y = -lnx - 2

20. Example
Find the domain of each function algebraically.