Logarithmic Functions
Exponential form
x  ay
Log Form

log a x  y;
Writing Equation in Log Form.
Write 32  9 in log form.
32  9

log 3 9  2
a  0 and x  0
Write 53  125 in log form.
53  125
Write 42 
42 
1
16

log 5125  3
1
in log form.
16

1
log 4    2
 16 
3
1 1
Write    in log form
2 8
3
1 1

  
2
8
 
1
log1/2    3
8
Write 3 64  4 in log form.
3
64  4
3
641  4
641/ 3  4

log 64 4  1/ 3
Writing Equation in Exponential Form
Write log 4 16  2 in exponential form
log 4 16  2

42  16
Write log8 1  0 in exponential form
log8 1  0
 80 = 1
Write in exponential form
log 2 16  4
24  16
Evaluate Log Expression
Evaluate log 3 9.
Solution:
Method 1: log 3 9  log 3  32   2
Method 2: log 3 9 
log10 (9)
2
log10 (3)
Evaluate log1/ 3 27.
Solution:
3
3
1
1
Method 1: log1/ 3 27  log1/ 3    3 Note:   = 27
3
 3
log10 (27)
Method 2: log1/ 3 9 
 3
log10 (1/ 3)
Evaluate log11 11.
Solution:
Method 1: log11 11  log11 111/ 2  =1/2
Method 2: log11 11 
log10
 11  1/ 2
log10 (11)
Evaluate 6log6 (11) .
Solution:
Log property: blogb ( x )  x
6log6 (11)  11
Evaluate 13log13 (211) .
Solution:
Log property: blogb ( x )  x
13log13 (211)  211
Evaluate eln(21) .
Solution:
Log property: blogb ( x )  x
eln(21)  eloge (21)  21
Evaluate eln(2 x1) .
Solution:
Log property: blogb ( x )  x
eln(2 x1)  eloge (2 x1)  2 x  1
Evaluate eln(5 x7) .
Solution:
Note: Log property: b logb ( x )  x
eln(5 x7)  eloge (5 x7)  5 x  7
Evaluate eln(9  3 x ) .
Solution:
Note: Log property: b logb ( x )  x
eln(93 x )  eloge (93 x )  9  3 x
Evaluate ln(21).
Solution:
Note: Ln = log e
where e = 2.718281828
ln(21) = log e (21) =3.044522438
Evaluate ln(455).
Solution:
Note: Ln = log e where e = 2.718281828
ln(455) = log e (455) =6.120297419
Evaluate ln(577).
Solution:
Note: Ln = log e where e = 2.718281828
ln(577) = log e (577) =6.357842267
Graphing Log Functions
Graph the function f ( x)  log 2 x.
f ( x)  log 2 x is defined when x  0.
f ( x)  log 2 x
x
0.1
1
2
4
8
log10 (0.1)
 3.32
log10 (2)
log10 (1)
log 2 1 
0
log10 (2)
log10 (2)
log 2  2  
1
log10 (2)
log10 (4)
log 2  4  
2
log10 (2)
log10 (8)
log 2  8  
3
log10 (2)
log 2 .1 
Point on Graph
(0.1, -3.32)
(1, 0)
(2, 1)
(4, 2)
(8, 3)
Graph the function f ( x)  log1/ 3 x.
Note: x > 0.
f ( x)  log1/ 3 x
x
0.1
1
1/3
1/9
1/27
log10 (0.1)
 2.095
log10 (1/ 3)
log10 (1)
log1/ 3 1 
0
log10 (1/ 3)
log10 (1/ 3)
log1/ 3 1/ 3 
1
log10 (1/ 3)
log10 (1/ 9)
log1/ 3 1/ 9  
2
log10 (1/ 3)
log10 (1/ 27)
log1/ 3 1/ 27  
3
log10 (1/ 3)
log1/ 3 .1 
Point on Graph
(0.1, 2.095)
(1, 0)
(1/3, 1)
(1/9, 4)
(1/27, 3)
Graph the function f ( x)  log3 ( x  2).
The function f(x) = log₃(x + 2) is defined when x+2 is greater than
0.
To figure out when x+2 is greater than 0, set x+2 > 0 and solve for
x:
x+2 > 0
x+2 -2 > 0 - 2
x > -2
Domain of the function f(x) = log₃(x + 2) is ( -2, ∞)
To graph the function f(x) = log(x + 2), plot the following points:
Note that f(x) = log₃(x+2) = log₁₀(x+2)/log₁₀(3) =
log₁₀(x+2)/log₁₀(3)
Point 1: (-1.9, -2.09590327428939)
when x = -1.9, y = log₃(-1.9+2) = log₁₀(-1.9+2)/log₁₀(3) = 2.09590327428939
Point 2: (-1, 0)
when x = -1, y = log₃(-1+2) = log₁₀(-1+2)/log₁₀(3) = 0
Point 3: (0, 0.630929753571458)
when x = 0, y = log₃(0+2) = log₁₀(0+2)/log₁₀(3) =
0.630929753571458
Point 4: (1, 1)
when x = 1, y = log₃(1+2) = log₁₀(1+2)/log₁₀(3) = 1
Point 5: (2, 1.26185950714292)
when x = 2, y = log₃(2+2) = log₁₀(2+2)/log₁₀(3) =
1.26185950714292
Graph log function with graphing calculator
f(x) = log₃(x+2) = log₁₀(x+2)/log₁₀(3) = log(x+2)/log(3)
(Input this function on graphing calculator; note log₁₀ is the same
as log.)
Graph the function f ( x)  ln(4  x).
The function f(x) = ln(4 - x) is defined when 4-x is greater than 0.
To figure out when 4-x is greater than 0, set 4-x > 0 and solve for
x:
4-x > 0
4-x-4 > 0-4
-x > -4
-x
-4
>
-1 -1
x<4
Domain of the function f(x) = ln(4 - x) is (-∞, 4)
To graph the function f(x) = ln(4 - x), plot the following points:
Point 1: (3.9, -2.30258509299405)
when x = 3.9, y = ln(4-3.9) = -2.30258509299405
Point 2: (3, 0)
when x = 3, y = ln(4-3) = 0
Point 3: (2, 0.693147180559945)
when x = 2, y = ln(4-2) = 0.693147180559945
Point 4: (1, 1.09861228866811)
when x = 1, y = ln(4-1) = 1.09861228866811
Point 5: (0, 1.38629436111989)
when x = 0, y = ln(4-0) = ln(4-0) = 1.38629436111989
Graph log function with graphing calculator
f(x) = ln(4-x)
Inverse Function of Log Function
Find the inverse function for f ( x)  log 2 x.
Write f ( x)  log 2 x as y  log 2 x.
Interchange x and y : x  log 2 y.
Solve for y :
2x  y
Inverse Function: y  2 x
Find the inverse function for f ( x)  log1/ 3 x.
Write f ( x)  log1/ 3 x as y  log1/ 3 x.
Interchange x and y : x  log1/ 3 y.
Solve for y :
(1/3) x  y
1
Inverse Function: y   
3
x