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Chapter 12: Problem 74

How could you convince someone that $$\log _{a} c \neq \log _{c} a ?$$

Short Answer

Expert verified
Since \(xy = 1\) and \(x eq y, \ \text{log}_a c eq \text{log}_c a\).

Step by step solution

01

Understand Logarithm Properties

First, recall that the logarithm \(\text{log}_b a\) tells us the power to which we must raise the base \(b\) to obtain the number \(a\). Meaning \(b^{\text{log}_b a} = a\).
02

Define Each Logarithm

Define \(x = \text{log}_a c\) and \(y = \text{log}_c a\). Therefore, we can write two equations based on our definitions: \(a^x = c\) and \(c^y = a\).
03

Analyze Potential for Equality

Substitute \(c\) from the first equation into the second equation: \((a^x)^y = a\). This simplifies to \(a^{xy} = a\). Hence, \(xy = 1\).
04

Verify Consistency

If \(xy = 1\), then \(y = \frac{1}{x}\). This means that the relationship between \(x\) and \(y\) is inverse and implies \(x eq y\).
05

Conclude the Proof

Since \(x = \text{log}_a c\) and \(y = \text{log}_c a\), and because \(x eq y\), it follows that \(\text{log}_a c eq \text{log}_c a\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm Properties
Logarithms have several fundamental properties that make them useful in solving equations and understanding mathematical relationships. A logarithm \(\text{log}_b a\), for instance, tells us the power to which the base \(b\) must be raised to obtain the number \(a\). In mathematical form, this is represented as \(b^{\text{log}_b a} = a\). This means if you know the base and the exponent, you can find the value, and vice versa.
Inverse Relationship
The inverse relationship between logarithms is crucial. In our exercise, we define \(x = \text{log}_a c\) and \(y = \text{log}_c a\). This sets up equations \(a^x = c\) and \(c^y = a\). Substituting \(c\) from the first equation into the second, we get \(a^{xy} = a\), which simplifies to \(xy = 1\). From this, we see that \(x\) and \(y\) are inversely related, meaning \(y = \frac{1}{x}\). Therefore, \(\text{log}_a c eq \text{log}_c a\), as \(x eq y\).
Logarithmic Equations
Understanding logarithmic equations is essential to solving many math problems. As shown in the exercise, we start with \(a^x = c\) and \(c^y = a\). These forms help us transform and analyze the equations. The equality \(a^{xy} = a\) leads us to \(xy = 1\), a crucial step proving the inequality of the logarithms. Hence, solving logarithmic equations often involves converting between exponential and logarithmic forms.
Proof by Contradiction
Proof by contradiction is a powerful method to show that something is true or false. In this case, we assumed \(\text{log}_a c = \text{log}_c a\) to check for consistency. Through analysis, we showed that this leads to a contradiction because it implies \(x\) equals \(y\), but our earlier steps proved \(x eq y\). Thus, we concluded that \(\text{log}_a c eq \text{log}_c a\), leveraging the contradiction in our assumptions to reach the final proof.

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Most popular questions from this chapter

The bacteria Escherichia coli (E. coli) are commonly found in the human bladder. Suppose that 3000 of the bacteria are present at time \(t=0 .\) Then \(t\) minutes later, the number of bacteria present is $$ N(t)=3000(2)^{t / 20} $$ If \(100,000,000\) bacteria accumulate, a bladder infection can occur. If, at 11: 00 A.M., a patient's bladder contains \(25,000 E\) coli bacteria, at what time can infection occur?

Express as an equivalent expression that is a single logarithm and, if possible, simplify. $$\log _{a}\left(x^{2}-4\right)-\log _{a}(x+2)$$

Simplify. $$ \log _{2}\left(\log _{2}\left(\log _{4} 256\right)\right) $$

Given \(\log _{b} 3=0.792 \text { and } \log _{b} 5=1.161\). If possible, use the properties of logarithms to calculate numerical values for each of the following. $$\log _{b} \frac{1}{3}$$

Rewrite each of the following as an equivalent logarithmic equation. Do not solve. $$ e^{3}=20.0855 $$
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