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THE NATURE OF GROWTH
Copyright © Cengage Learning. All rights reserved.
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Copyright © Cengage Learning. All rights reserved.
10.2 Logarithmic Equations
3
Fundamental Properties
4
Fundamental Properties
With a little earthquake research, you could find information on the Richter scale, which was developed by Gütenberg and Richter.
The formula relating the energy E (in ergs) to the magnitude of the earthquake M is given by
This equation is called a logarithmic equation, and the topic of this section is to solve such equations.
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Fundamental Properties
To answer the question, we first solve for log E, and then we use the definition of logarithm to write this as an exponential equation.
1.5M = log E – 11.8
1.5M + 11.8 = log E
101.5M + 11.8 = E
Given equation
Multiply both sides by 1.5.
Add 11.8 to both sides.
Definition of logarithm
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Fundamental Properties
We can now answer the question. Since M = 7.6,
E = 101.5(7.6) + 11.8
1.58 1023
However, to solve certain logarithmic equations, we must first develop some properties of logarithms.
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Fundamental Properties
We begin with two fundamental properties of logarithms. If you understand the definition of logarithm, you can see that these two properties are self-evident, so we call these the Grant’s tomb properties of logarithms.
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Example 1 – Evaluate a logarithmic expression
Evaluate the given expressions.
a. log 103
b. ln e2
c. 10log 8.3
d. eln 4.5
e. log8 84.2
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Example 1 – Solution
a. log 103 = 3
b. ln e2 = 2
c. 10log 8.3 = 8.3
d. eln 4.5 = 4.5
e. log8 84.2 = 4.2
log 103 is the exponent on 10 that gives 103; obviously it is 3.
ln e2 is the exponent on e that gives e2; obviously it is 2.
Consider log 8.3; what is this? It is the exponent on a base 10 that gives the answer 8.3.
Consider ln 4.5; what is this? It is the exponent on a base e that gives the answer 4.5.
What is the exponent on 8 that gives 84.2? Obviously, it is 4.2.
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Logarithmic Equations
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Logarithmic Equations
A logarithmic equation is an equation for which there is a logarithm on one or both sides.
The key to solving logarithmic equations is the following theorem, which we will call the log of both sides theorem.
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Laws of Logarithms
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Laws of Logarithms
To simplify logarithmic expressions, we remember that a logarithm is an exponent and the laws of exponents correspond to the laws of logarithms.
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Laws of Logarithms
The proofs of these laws of logarithms are easy. The additive law of logarithms comes from the additive law of exponents:
bxby = bx + y
Let A = bx and B = by, so that AB = bx + y. Then, from the definition of logarithm, these three equations are equivalent to
x = logb A, y = logb B, and x + y = logb(AB)
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Laws of Logarithms
Therefore, by putting these pieces together, we have
logb(AB) = x + y = logb A + logb B
Similarly, for the subtractive law of logarithms,
Subtractive law of exponents
Definition of logarithm
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Laws of Logarithms
The proof of the multiplicative law of logarithms follows from the multiplicative law of exponents and you are asked to do this in the problem set. We can also prove this multiplicative law by using the additive law of logarithms for p a positive integer:
logb Ap = logb (A A A A)
= logb A + logb A + logb A + + logb A
Since x = logb A and y = logb B
Definition of Ap
Additive law of logarithms
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Laws of Logarithms
When logarithms were used for calculations, the laws oflogarithms were used to expand an expression such as
log .
Calculators have made such problems obsolete.
Today, logarithms are important in solving equations, and the procedure for solving logarithmic equations requires that we take an algebraic expression involving logarithms and write it as a single logarithm.
We might call this contracting a logarithmic expression.
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Example 3 – Use logarithmic properties to contract
Write each statement as a single logarithm.
a. log x + 5 log y – log z
b. log2 3x – 2 log2 x + log2(x + 3)
Solution:
a. log x + 5 log y – log z = log x + log y 5 – log z
= log xy 5 – log z
=
Third law
First law
Second law
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Example 3 – Solution
b.
cont’d
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Example 6 – Prove change of base theorem
Prove:
Solution:
Let y = loga x.
ay = x
logb ay = logb x
y logb a = logb x
Definition of logarithm
Log of both sides theorem
Third law of logarithms
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Example 6 – Solution
Thus, by substitution, .
cont’d
Divide both sides by logb a (logb a 0).
