18 Logarithmic Functions
Concepts:
– Logarithms as Functions – Logarithms as Exponent Pickers – Inverse Relationship between Logarithmic and Exponential Functions. – The Common Logarithm ∗ Definition and Graphs ∗ Exponential Notation vs. Logarithmic Notation ∗ Evaluating Common Logarithms – The Natural Logarithm ∗ Definition and Graphs ∗ Exponential Notation vs. Logarithmic Notation ∗ Evaluating Common Logarithms – Logarithms with Different Bases ∗ Definition and Graphs ∗ Exponential Notation vs. Logarithmic Notation ∗ Evaluating Common Logarithms
(Section 3.3)
18.1 Logarithms
Exponential functions are one-to-one functions. Consequently, each exponential function has an inverse function. Recall that inverse functions are useful because they undo operations. Addition and Subtraction are inverse operations. Multiplication and Division are inverse operations. Similarly, exponentiation and taking the logarithm are inverse operations. These operations undo each other. Why might you want to undo exponentiation? Suppose you want to solve the following equation. 10x = 3
What is happening to x?
1 How do we undo this? Taking the xth root is not a reasonable solution. This would lead to: √ 10 = x 3
1 10 = 3 x 1 This is even worse than before. We now have x as an exponent. What we need is something to pull x out of the exponent place and put it on the ground, in a manner of speaking. Logarithms are the answer. One Calculus teacher was fond of saying that logarithms are “exponent pickers.” Recall that the name of a function does not need to be a single letter. Up to this point in the course, we have been very lazy and unimaginative in naming our functions. It is not terribly useful to call a functions f or g, but these names are sufficient for simple examples. Moreover, we have used f and g to mean lots of different functions. But some functions occur so regularly that it makes more sense to give them permanent names that are a bit more descriptive. This is the case with logarithms. Logarithms will have names like log,
log2, log3, and ln. Because these are functions, they have inputs and these inputs are placed in parentheses next to the function name. For example log(x), is the output of the function named log when x is the input of the function.
18.1.1 The Common Logarithm
The common logarithm is the inverse function of f(x) = 10x. The name of the common
logarithm function is either log10 (said “log base 10”) or log for short. Example 18.1 (Common Log Graph)
• Sketch the graphs of y = f(x) = 10x and y = g(x) = log(x) on the same coordinate system.
• What is the domain of log?
2 Recall that x and y trade places in inverse functions. This leads to the following definition for the common logarithm function. Definition 18.2 (Common Logs) Let x and y be real numbers with x > 0. Then
log(x) = y if and only if 10y = x.
In other words, log(x) chooses (or picks) the exponent to which 10 must be raised to produce x. Example 18.3 (Common Logs) Evaluate each of the following.
• log(100)
• log(109)
1 • log 1000
1 • log 1020
√ • log( 3 100)
Example 18.4 (Exponential Notation and Logarithmic Notation) Convert the logarithmic statement to an exponential statement. log(10, 000) = 4
Example 18.5 (Exponential Notation and Logarithmic Notation) Convert the exponential statement to a logarithmic statement. 1 10−3 = 1, 000
3 18.1.2 The Natural Logarithm
As we mentioned before, the function f(x) = ex is especially important in Calculus, Business Applications, Engineering Applications, and Biological Applications. The inverse function of f is ln(x). This is called the natural logarithm function.
Example 18.6 (Natural Log Graph)
• Sketch the graphs of y = f(x) = ex and y = g(x) = ln(x) on the same coordinate system.
• What is the domain of ln?
Recall that x and y trade places in inverse functions. This leads to the following definition for the common logarithm function. Definition 18.7 (Natural Logs) Let x and y be real numbers with x > 0. Then
ln(x) = y if and only if ey = x.
In other words, the ln(x) chooses (or picks) the exponent to which e must be raised to produce x.
Example 18.8 (Natural Logs) Evaluate each of the following.
• ln(e5)
! r 1 • ln 6 e11
4 Example 18.9 (Exponential Notation and Logarithmic Notation) Convert the logarithmic statement to an exponential statement.
ln(1) = 0
Example 18.10 (Exponential Notation and Logarithmic Notation) Convert the exponential statement to a logarithmic statement.
e2 ≈ 7.389
Example 18.11 (Logarithm Domain) Find the domain of f(x) = ln(x2 − 3x − 10).
18.1.3 Logarithms with Different Bases
The common logarithm (log) is sometimes called “log base 10” and can also be written log10. The natural logarithm (ln) is sometimes called “log base e” and can also be written loge. In fact, there is a logarithm associated with any positive base a that is not equal to 1. Definition 18.12 (Logarithms with Different Bases) Let x and y be real numbers with x > 0. Let a be a positive real number that is not equal to 1. Then
y loga(x) = y if and only if a = x.
In other words, the loga(x) picks the exponent to which a must be raised to produce x.
5 Example 18.13 (Logartithms with Different Bases) Evaluate each of the following.
• log2(16)
1 • log 3 81
• log 1 (16) 2
Example 18.14 (Exponential Notation and Logarithmic Notation) Convert the exponential statement to a logarithmic statement.
53 = 125
Example 18.15 (Logarithm Domain)
Find the domain of f(x) = log7(2 − 5x).
6 18.1.4 Properties of Logarithms
Each property of logarithms is derived from the definition of the logarithm and/or a property of exponents. Property 18.16 • log(1) =
• ln(1) =
• loga(1) =
Property 18.17 • log(10) =
• ln(e) =
• loga(a) =
Property 18.18 • log(10x) =
• ln(ex) =
x • loga(a ) =
Property 18.19 • 10log(x) =
• eln(x) =
• aloga(x) =
7 Example 18.20
• Simplify ex ln(2).
• Rewrite 5x as e to a power.
Example 18.21 Evaluate log(105 ∗ 103).
Property 18.22 (Product Law for Logarithms) For all u > 0 and v > 0
• log(uv) = log(u) + log(v)
• ln(uv) = ln(u) + ln(v)
• loga(uv) = loga(u) + loga(v) Proof:
8 Property 18.23 (Quotient Law for Logarithms) For all u > 0 and v > 0 u • log = log(u) − log(v) v u • ln = ln(u) − ln(v) v u • log = log (u) − log (v) a v a a
Example 18.24 xy Use the properties of logarithms to express ln as a sum and or difference of three z logarithms.
Example 18.25 Use the properties of logarithms to write the expression using the fewest number of logarithms possible. log(x2 + 2) + log(x) − log(y) + log(w) log(z)
Property 18.26 (Power Law for Logarithms) For all u > 0 and all k
• log(uk) = k log(u)
• ln(uk) = k ln(u)
k • loga(u ) = k loga(u)
Proof:
9 Example 18.27 x3 Use the properties of logarithms to express log √ in terms of log(x), log(y), and log(z). y z
10 Example 18.28 Use the properties of logarithms to write the expression using the fewest number of logarithms possible. log(x2) − 2 log(y) − 3 log(z)
Property 18.29 (Change of Base) If a, b, c > 0 and neither a nor b equals 1, then
logb(c) loga(c) = . logb(a)
Example 18.30 (Change of Base)
Use your calculator to approximate log5(67).
Example 18.31 (Earthquakes) You should read Example 10 in Section 5.4 of your textbook about the Richter scale. It is interesting.
11 18.2 Solving Exponential and Logarithmic Equations
Example 18.32 Solve. log(x + 5) = 3
Example 18.33 Solve.
log8(x − 5) + log8(x + 2) = 1
Example 18.34 Solve. ex+2 = 5
12 Example 18.35 Solve. 2x − 7 = −1 3
Example 18.36 Solve. 2x−5 = 32−2x
Example 18.37 Joni invests 1000 at an interest rate of 5% compounded monthly. When will the value of Joni’s investment reach $2500?
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