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Elementary Functions
Part 3, Exponential Functions & Logarithms Lecture 3.3a, Logarithms: Basic Properties
Dr. Ken W. Smith
Sam Houston State University
2013
Smith (SHSU) Elementary Functions 2013 1 / 29
The logarithm as an inverse function
In this section we concentrate on understanding the logarithm function. If the logarithm is understood as the inverse of the exponential function, then the properties of logarithms will naturally follow from our understanding of exponents.
The meaning of the logarithm. The logarithmic function g(x) = logb(x) is the inverse of the exponential function f (x) = bx.
The meaning of y = logb(x) is by^ = x.
The expression by^ = x is the “exponential form” for the logarithm y = logb(x).
The positive constant b is called the base (of the logarithm.)
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Logarithms
Some worked exercises. Write each of the following logarithms in exponential form and then use that exponential form to solve for x. 1 log 2 (8) = x Solution. The exponential form is 2 x^ = 8. Since 23 = 8 the answer is x = 3. 2 log 2 (2^47 ) = x Solution. The exponential form is 2 x^ = 2^47. So x = 47. 3 log 2 ( 12 ) = x Solution. The exponential form is 2 x^ = 12. Since 2 −^1 =
the answer is x = − 1.
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Logarithms
4 log 2 ( 18 ) = x Solution. The exponential form is 2 x^ = 18. Since 2 −^3 = 18 the answer is x = − 3. 5 log 2 ( 3
- = x Solution. The exponential form is 2 x^ = 3
2 = 2^1 /^3. So x = 1/ 3.
Notice how we use the exponential form in each problem!
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The graph of a logarithm function
The graph of y = 2x^ was drawn in an earlier lecture (see below.)
The graph of the inverse function y = log 2 x is obtained by reflecting the graph of y = 2x^ across the line y = x. Smith (SHSU) Elementary Functions 2013 5 / 29
The graph of a logarithm function
The graph of y = log 2 x :
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The graph of a logarithm function
If we draw them together, we have the picture below.
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The graph of a logarithm function
The graph of the exponential function y = 2x:
The graph of the logarithmic functionSmith (SHSU) Elementary Functions y = log 2 x: 2013 8 / 29
Logarithms
What happens when we divide two terms with a common base?
bx by^
= bx−y^ (3)
When we do division, we subtract exponents. So, in the language of logarithms, we have the quotient property, “the exponent in a quotient is the difference of the two exponents”:
logb(
M
N
) = logb M − logb N (4)
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Logarithms
A third important property of exponents: when we raise a term like bx^ to a power, we multiply exponents.
(bx)c^ = bxc^ (5)
In our “logarithm language” (thinking of M as bx) we have the exponent property
logb(M c) = c logb M (6)
Each of these three properties is merely a restatement, in the language of logarithms, of a property of exponents.
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Logarithms
We review the three basic logarithm rules we have developed so far.
Product Property of Logarithms: logb(M N ) = logb M + logb N
Quotient Property of Logarithms:
logb(
M
N
) = logb M − logb N
Exponent Property of Logarithms:
logb(M c) = c logb M
Each of these properties is a restatement, in the language of logarithms, of a property of exponents. Smith (SHSU) Elementary Functions 2013 15 / 29
Exponential Functions
In the next presentation, we develop several more properties of logarithms.
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Elementary Functions
Part 3, Exponential Functions & Logarithms Lecture 3.3b, Logarithms: Basic Properties, Continued
Dr. Ken W. Smith
Sam Houston State University
2013
Smith (SHSU) Elementary Functions 2013 17 / 29
Logarithms
We review the three basic logarithm rules we have developed so far.
Product Property of Logarithms: logb(M N ) = logb M + logb N
Quotient Property of Logarithms:
logb(
M
N
) = logb M − logb N
Exponent Property of Logarithms:
logb(M c) = c logb M
Each of these three properties is merely a restatement of a property of exponents. Smith (SHSU) Elementary Functions 2013 18 / 29
Changing the base
Suppose we want to change the base of our logarithm. This often occurs when we want to use a “good” base like e on a problem which began with a different base. Suppose we want to work with base c but our problem began with base b:
y = logb x. Rewrite this in exponential form: by^ = x. Now take the log of both sides of the equation. If we want to work in base c then let us apply logc() to both sides of our equation. logc(by) = logc(x). Now we use the exponent property pulling the exponent y outside the logarithm: y logc(b) = logc(x). Solve for y: y =
logc x Smith (SHSU) Elementary Functions. 2013 19 / 29
Changing the base
We began with y = logb x. We rewrote this as y =
logc x logc b
So y, which was originally equal to logb x is now
logb x =
logc x logc b
Let’s call this the “change of base” equation or “change of base” property.
One way to remember this is to note that on the left side of the equal sign (logb x), b is lower than x. Then on the right side of the equal sign ( log log^ xb ), b is still lower than x!
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Six properties of logarithms
4 The Change of Base Property
logb x =
logc x logc b
5 Inverse Property # logb bx^ = x
6 Inverse Property # blogb^ x^ = x
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More on the logarithm as an inverse function
If we understand the logarithm as the inverse of the exponential function then we are prepared to find the inverse of a variety of functions. Here are some examples. Find the inverse function of: 1 f (x) = ex
2 . 2 f (x) = ex^2 −^5. 3 f (x) = 5 + ex. 4 f (x) = log 2 (x + 2) + 2.
Solutions 1 To find the inverse of f (x) = ex^2 set y = ex^2 and swap inputs and outputs x = ey
2 . Take the natural logarithm of both sides ln x = y^2 and solve for y by taking square roots of both sides √ ln x = y.
So one inverse function is f −^1 (x) =
ln x.
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More on the logarithm as an inverse function
2 To find the inverse of y = ex
(^2) − 5 we swap letters so that x = ey
(^2) − 5 , take natural logs of both sides
ln x = y^2 − 5 ,
add 5 and take square roots so that
f −^1 (x) =
ln x + 5.
3 To find the inverse of y = 5 + ex^ we swap variables, subtract 5 from both sides and then take the natural log to get ln(x − 5) = y. So
f −^1 (x) = ln(x − 5).
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More on the logarithm as an inverse function
4 To find the inverse of f (x) = log 2 (x + 2) + 2 we write y = log 2 (x + 2) + 2, change variables (to indicate that we are swapping inputs and outputs) x = log 2 (y + 2) + 2, and subtract 2 from both sides x − 2 = log 2 (y + 2). At this point it is best to write this logarithmic equation in exponential form. 2 x−^2 = y + 2. Subtract 2 from both sides 2 x−^2 − 2 = y and then write out our answer using inverse function notation.
f −^1 (x) = 2x−^2 − 2 Smith (SHSU) Elementary Functions 2013 28 / 29
Exponential Functions
In the next series of lectures, we apply properties of logarithms.
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