6.4 Logarithmic Functions
x
A problem without a solution with what we have done so far: Find the inverse of y = b
We need something new! The inverse of an exponential function is a logarithm.
The answer to logb x is what you write in the box: b = x
Example 1: Evaluate without using a calculator. A.
log10 100
B. log 3
1
3
1
49
LOGARITHMIC DEFINITION: y = log b x only if x = b y where b >0 but is not 1; x is > 0 and y is real.
C. log5 125
D.
log 7
y = log b x is read “y equals log base b of x”. Write the b smaller and lower than the rest of the expression.
Exponential Form
number = base
power
Logarithmic Form
Exponential Form
power = logbase number
number = base
Logarithmic Form
power
power = logbase number
4
52 = 25
10 = 10000
log 3 9 = 2
log5
43 = 64
1
= −2
25
dg = k
Natural Logarithm: If the logarithm is base e, we can abbreviate it, as in log e 5 = ln 5
Common Logarithm: If the logarithm is base 10, we do not need to write the 10 in the base. log10 5 = log 5
There are keys on your calculator for both the natural LN and common logarithm LOG .
Example 3: Use your calculator to approximate the expression to four decimal places where applicable.
A. log 5
B. ln 5
C. log 100 D. log (-10) E. ln 0.42
F. ln 1 G. log 10
H. log(7) +1
2 + ln 5
ln 3
I.
We can find the domain of a log function two ways.
A) Keep the argument positive: for y = 2 log 5 ( 3 x − 7 ) + 10 the argument must be positive, as in 3x – 7 > 0.
B) Draw the graph using transformations as is done below.
Example 4: Graph the exponential function y = 5 x below. Then graph its inverse logarithmic function y = log 5 x .
x
y = 5x
x
x
Exponential Function: Domain:
Logarithmic Function: Domain:
Range:
Range:
Asymptote:
Asymptote:
y = log 5 x
Example 5: Graph the function y = log 5 ( x − 3 ) . First find points on y = log 5 ( x ) and then do the transformation.
x
x
y = 5x
x
y = log 5 ( x )
y = log 5 ( x − 3 )
x
y = log 5 ( x − 3 )
Domain:
Range:
Asymptote:
Example 6: Graph the function y = ln ( x ) + 1 . First find points on y = ln ( x ) and then do the transformation.
x
y = ex
x
x
y = ln ( x )
y = ln ( x ) + 1
x
y = ln ( x ) + 1
Domain:
Range:
Asymptote:
Example 7: Graph the function y = − log 2  x  − 4 . First find points on y = log 2 ( x ) and then do the transformation.
3
x
x
x
y = − log 2   − 4
3
Domain:
y = log 2 ( x )
Range:
x
 x
y = − log 2  
3
x
Asymptote:
x
y = − log 2   − 4
3