1. Logarithmic Functions
Logarithmic Functions and Models
Logarithmic Functions
Logarithmic Functions and Models
Logarithmic Functions
We investigate logarithms, logarithmic functions and their connection to exponential functions.
Logarithmic Functions
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2. Logarithmic Functions
Logarithmic Functions and Models
What Is a Logarithm ?
The base-a logarithm of x is the power of a that yields x, written
The base-a logarithm of x is the inverse function of the base-a exponential function, that is
log a x
What Is a Logarithm ?
We define a logarithm as a power of a base number and connect it directly to a corresponding exponential function with the same base number – that is, the logarithm is the exponent for the corresponding exponential base.
log a (ax) = x
log a x a = x
and
See Logarithms
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Logarithmic Functions
Logarithmic Functions
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3. Logarithmic Functions
Logarithmic Functions and Models
Consider the exponential function
f(x) = ax , 0 < a ≠ 1
We can show that f(x) is a 1–1 function
So f(x) does have an inverse function f –1(x)
Since we showed that
we call this inverse the logarithm function
with base a
f –1 (f(x)) = f –1 (ax )
= a
loga x
= x
Logarithmic Functions
Here we make the connection that loga x and ax are inverses of each other, so each of them is a 1-1 function.
We note that the domain of one is also the range of the other. The domain of ax is the set of real numbers R and its range is the set of all positive real numbers, (0, ∞).
The domain of the logarithm function is the set of all positive real numbers, (0, ∞) and its range is the set of all real numbers, R.
log a ay = log a x
= y
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Logarithmic Functions
Logarithmic Functions
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4. Logarithmic Functions
Logarithmic Functions and Models
Consider the exponential function
Since log a x is an inverse of a x then
log a (ax) = x
f(x) = a x, 0 < a ≠ 1
log a = x
Logarithmic Functions
Here we make the connection that loga x and ax are inverses of each other, so each of them is a 1-1 function.
We note that the domain of one is also the range of the other. The domain of ax is the set of real numbers R and its range is the set of all positive real numbers, (0, ∞).
The domain of the logarithm function is the set of all positive real numbers, (0, ∞) and its range is the set of all real numbers, R.
Questions:
What are the domain and range of a x ?
What are the domain and range of log a x ?
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Logarithmic Functions
Logarithmic Functions
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5. Logarithmic Functions
Logarithmic Functions and Models
A Second Look
Graphs of f(x) = ax and
f –1(x) = log a x are mirror
images with respect
to the line y = x
Note that y = f –1(x)
if and only if f(y) = a y = x
y = loga x iffi a y = x x y
f(x) = ax
, for a > 1
y = x
(1, a) ●
f–1(x) = logax
(0, 1) ● ●
(a, 1) ●
(1, 0)
Logarithmic Functions: A Second Look
We look at the graphs of f(x) = ax and f–1(x) = logax and verify the domains and ranges of the functions. We note that, as with all inverse function pairs, the graphs are symmetric with respect to the line y = x.
As the illustration shows, each point (x, y) on one graph has a corresponding point (y, x) on the other graph. This is exactly the concept of inverse function. The points (x, y) and (y, x) can also be viewed simply as ordered pairs in the sets of ordered pairs that are the functions.
Remember:
loga x is the power of a that yields x
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Logarithmic Functions
Logarithmic Functions
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6. Logarithmic Functions
Logarithmic Functions and Models
A Third Look
log a x = y means a y = x
What if x = a ? x y
f(x) = ax
, for a > 1
y = x
(1, a)
Then ●
f–1(x) = logax
log a a = y and a y = a
(0, 1) ● ●
(a, 1) ●
So for any base a ,
(1, 0)
with a > 0, a ≠ 1
Logarithmic Functions: A Second Look
We look at the graphs of f(x) = ax and f–1(x) = logax and verify the domains and ranges of the functions. We note that, as with all inverse function pairs, the graphs are symmetric with respect to the line y = x.
As the illustration shows, each point (x, y) on one graph has a corresponding point (y, x) on the other graph. This is exactly the concept of inverse function. The points (x, y) and (y, x) can also be viewed simply as ordered pairs in the sets of ordered pairs that are the functions.
log a a = 1
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Logarithmic Functions
Logarithmic Functions
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7. Logarithmic Functions
Logarithmic Functions and Models
A Third Look
log a x = y means a y = x
What if x = 1 ? x y
f(x) = ax
, for a > 1
y = x
(1, a)
Then ●
f–1(x) = logax
a y = 1 and so y = 0
(0, 1) ● ●
(a, 1)
WHY ? ●
(1, 0)
So for any base a ,
Logarithmic Functions: A Second Look
We look at the graphs of f(x) = ax and f–1(x) = logax and verify the domains and ranges of the functions. We note that, as with all inverse function pairs, the graphs are symmetric with respect to the line y = x.
As the illustration shows, each point (x, y) on one graph has a corresponding point (y, x) on the other graph. This is exactly the concept of inverse function. The points (x, y) and (y, x) can also be viewed simply as ordered pairs in the sets of ordered pairs that are the functions.
with a > 0, a ≠ 1
log a 1 = 0
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Logarithmic Functions
Logarithmic Functions
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8. Logarithmic Functions
Logarithmic Functions and Models x y
f(x) = ax,
A Fourth Look
What if 0 < a < 1 ?
Then what is f –1(x) ?
Are the graphs still
symmetric with
respect to line y = x ?
YES !
for a < 1
y = x
(a, 1) ●
(0, 1)
(1, a) ●
(1, 0)
Logarithmic Functions: A Fourth Look
This slide explores what happens when the base a is less than 1. The exponential growth function ax becomes an exponential decay function. The reflection of its graph through the line y = x is the graph of its inverse function, logax.
Clearly the two graphs intersect on the line y = x. The basic facts about both exponential and logarithmic functions still hold true:
The log of the base is always 1
The log of 1 is always 0
The exponential of 1 is always the base
The exponential of 0 is always 1
Since these facts are true for any base, they are certainly true for base a, where 0 < a < 1.
Having established that the graphs intersect on the line y = x the question we might ask is: for what value of x is this true? That is, can we solve the equation logax = ax
for x? We shall explore this question shortly.
f–1(x) = logax
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Logarithmic Functions
Logarithmic Functions
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9. Logarithmic Functions
Logarithmic Functions and Models x y
f(x) = ax,
A Fourth Look
Is it still true that
f(1) = a and f –1(x) = 1 ?
YES !
Is it possible that
log a x = a x ?
on the line y = x
for a < 1
y = x
(a, 1) ● ● ●
(0, 1) ●
(1, a) ●
(1, 0)
Logarithmic Functions: A Fourth Look
This slide explores what happens when the base a is less than 1. The exponential growth function ax becomes an exponential decay function. The reflection of its graph through the line y = x is the graph of its inverse function, logax.
Clearly the two graphs intersect on the line y = x. The basic facts about both exponential and logarithmic functions still hold true:
The log of the base is always 1
The log of 1 is always 0
The exponential of 1 is always the base
The exponential of 0 is always 1
Since these facts are true for any base, they are certainly true for base a, where 0 < a < 1.
Having established that the graphs intersect on the line y = x the question we might ask is: for what value of x is this true? That is, can we solve the equation logax = ax
for x? We shall explore this question shortly.
f–1(x) = logax
For what x does log a x = a x ?
Question:
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Logarithmic Functions
Logarithmic Functions
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10. Logarithmic Functions
Logarithmic Functions and Models
Example
Let a = ½
Then f(x) = ax
Then what is f–1(x) ?
f –1(x) = log a x = log½ x
It is still true that
f(1) = (½)1 = ½
and f–1(½) = log½(½) = 1
Where do the graphs intersect?
y = (½)x = log½x = x
f(x) = ax x y
for a = ½
y = x
= (½)x
(½,1) ● ●
(0,1) ● ●
(1,½) ●
(1,0)
Logarithmic Functions: A Fourth Look Example
This slide explores what happens when the base is less than 1. The exponential growth function f(x) = ax becomes the exponential decay function f(x) = (½)x for a = ½ < 1. The inverse of this function is the logarithmic function f–1(x) = log½x, shown in the illustration.
Note that in this case both functions are decreasing functions that intersect the line y = x. This means that on that line the two function values are identical to the value of x chosen in each case. Since the inverse function must carry the value of y produced by one function back to the original value of x chosen for the other, the values of y must be identical when y = x.
To determine where f(x) = f –1(x) we must employ graphical and/or numerical methods, since there is no solution for logax = ax in closed form (i.e., no symbolic solution). By plotting the graphs of f(x), f–1(x), and y = x we can use a graphing calculator to trace a point along one of the graphs, say y = x, until we see that y = x. This should occur when y = x ≈
If you have a TI graphing calculator, setting the window size to [–.5, 2, 1] by [–.5, 2, 1] makes this fairly obvious.
f –1(x) = log1/2 x
WHY ?
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Logarithmic Functions
Logarithmic Functions
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11. Exponent / Log Comparisons
Logarithmic Functions and Models
Compare ax with loga x
Exponential Inverse Function
102 = 100
34 = 81
(½)–5 = 32
25 = 32
5–3 = 1/125
e3 ≈
log10100 = 2
log3 81 = 4
log1/2 32 = –5
Exponential - Logarithmic Comparisons
The Common Logarithm (base 10) is also known as the Briggsian logarithm after Henry Briggs (1561 – 1639) of Gresham College, London who computed the first table of common logarithms.
The Natural, or Naperian, Logarithm was invented by John Napier (1550 – 1617) Scottish laird (Baron of Murchiston) in a slightly different form. It was Napier who named this value the logarithm from the Greek words logos (ratio) and arithmos (number).
Napier did not think in terms of a base but in terms of a geometric sequence based on the numbers 1 – 10 – 7 . In practice the Naperian log L of a number N was given by the expression
Thus Naper’s logarithm of 107 is 0 and his logarithm of 107(1 – 1/107) is 1. Dividing his numbers by 107 yields very nearly a system of logarithms with base 1/e. The ratios in his system were geometric ratios of proportional distances measured along a pair lines. The modern version of the natural logarithm uses base e where The variable n can be replaced by 1/x as x approaches 0 , that is,
log2 32 = 5
log5 (1/125) = –3
loge( )
≈ 3
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Logarithmic Functions
e as n 
e as x
Logarithmic Functions
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12. Exponent / Log Comparisons
Logarithmic Functions and Models
Compare Logarithms
Common Logarithm
Base 10
Log10 x = Log x
Natural Logarithm
Base e
Loge x = ln x
Exponential - Logarithmic Comparisons
The Common Logarithm (base 10) is also known as the Briggsian logarithm after Henry Briggs (1561 – 1639) of Gresham College, London who computed the first table of common logarithms.
The Natural, or Naperian, Logarithm was invented by John Napier (1550 – 1617) Scottish laird (Baron of Murchiston) in a slightly different form. It was Napier who named this value the logarithm from the Greek words logos (ratio) and arithmos (number).
Napier did not think in terms of a base but in terms of a geometric sequence based on the numbers 1 – 10 – 7 . In practice the Naperian log L of a number N was given by the expression
Thus Naper’s logarithm of 107 is 0 and his logarithm of 107(1 – 1/107) is 1. Dividing his numbers by 107 yields very nearly a system of logarithms with base 1/e. The ratios in his system were geometric ratios of proportional distances measured along a pair lines. The modern version of the natural logarithm uses base e where The variable n can be replaced by 1/x as x approaches 0 , that is,
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Logarithmic Functions
e as n 
e as x
Logarithmic Functions
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13. One-to-One Property Review
Logarithmic Functions and Models
Since a x and loga x are 1-1
1. If x = y then ax = ay
2. If ax = ay then x = y
3. If x = y then loga x = loga y
4. If loga x = loga y then x = y
These facts can be used to solve equations
Solve: 10x – 1 = 100
10x – 1 = 102
x – 1 = 2
x = 3
One-to-One Property Review
The salient feature to observe about 1-1 functions is that f(x) = f(y) if and only if x = y. The illustration shows what this means in terms of the exponential and logarithm functions.
We can use the fact that both the exponential and the logarithm functions are 1-1 to solve exponential and logarithm equations.
In the example we can move the variable from the exponent to a multiplier by using the logarithm to act on each side of the equation. This helps us form a linear equation in which we isolate x. By computing log 4 and log 10, in the example, we can solve for x. Note that log 10 = 1, since log without an explicit base is by convention log10. Solving for x gives us x = 3 { }
Solution set :
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Logarithmic Functions
Logarithmic Functions
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14. Logarithmic Functions and Models
Think about it !
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Logarithmic Functions
Logarithmic Functions
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