The table shows a student's proof of the quotient rule for logarithms.
Let [tex]$M =b^x$[/tex] and [tex]$N =b^y$[/tex] for some real numbers [tex]$x$[/tex] and [tex]$y$[/tex].
| | Step | Reason |
|---|---------------------------------------|--------------------------------------------------------------|
| 1 | [tex]$\log _b\left(\frac{M}{N}\right)$[/tex] | Given |
| 2 | [tex]$=\log _b\left(\frac{b^X}{b^Y}\right)$[/tex] | Substitution |
| 3 | [tex]$=\log _b\left(b^x\right)-\log _b\left(b^y\right)$[/tex] | Properties of logarithms |
| 4 | [tex]$=x-y$[/tex] | Logarithm property [tex]$\log _b\left(b^9\right)=c$[/tex] |
| 5 | [tex]$=\log _b(M)-\log _b(N)$[/tex] | Substitution |
What is the error in the proof?
A. The error is in step 3. The reason should be the quotient property of exponents.
B. The error is in step 3. You cannot use a property of logarithms to prove that same property.
Answer (1)
The proof attempts to show the quotient rule for logarithms.
The error occurs in Step 3, where the logarithm is split before applying the quotient rule of exponents.
The correct approach would be to simplify the fraction inside the logarithm first using the quotient rule of exponents: b y b x = b x − y .
Therefore, the error is in Step 3, and the reason should be the quotient property of exponents. The error is in step 3. The reason should be the quotient property of exponents.
Explanation
Analyzing the Proof We are given a proof of the quotient rule for logarithms and asked to identify the error. Let's analyze the steps.
Step 1 Step 1 states lo g b ( N M ) which is the given expression.
Step 2 Step 2 substitutes M with b x and N with b y , so we have lo g b ( b y b x ) . This is a valid substitution based on the initial definitions.
Step 3 Step 3 rewrites the expression as lo g b ( b x ) − lo g b ( b y ) . This step is incorrect. The correct procedure would be to first simplify the fraction inside the logarithm using the quotient rule of exponents, which states that b y b x = b x − y . Then, we would have lo g b ( b x − y ) .
Step 4 Step 4 uses the logarithm property lo g b ( b c ) = c to simplify lo g b ( b x ) − lo g b ( b y ) to x − y . This step is correct, assuming the previous step was correct.
Step 5 Step 5 substitutes back x with lo g b ( M ) and y with lo g b ( N ) , resulting in lo g b ( M ) − lo g b ( N ) . This step is correct, assuming all previous steps were correct.
Conclusion The error is in Step 3. Instead of separating the logarithm of a quotient into the difference of logarithms, the quotient rule of exponents should have been applied first.
Examples
The quotient rule of logarithms is useful in many fields, such as calculating the magnitude of earthquakes on the Richter scale or simplifying complex expressions in physics and engineering. For example, if you are analyzing the signal strength ratio in decibels, you might use the quotient rule to simplify the logarithmic expression representing the ratio.
