Question
Product Rule of Logarithms proof and Quotient Rule comparison
Answers
GPT-4o mini
The product rule and quotient rule of logarithms are fundamental properties that rest on the definitions of logarithms and exponents. Let’s go through the proof of each and then highlight how they compare.
### Product Rule of Logarithms
**Statement**:
If \( a > 0 \) and \( b > 0 \), then:
\[
\log_b (a \cdot c) = \log_b a + \log_b c
\]
**Proof**:
According to the definition of a logarithm, we have:
\[
\log_b a = x \implies b^x = a
\]
\[
\log_b c = y \implies b^y = c
\]
Now, we want to find \( \log_b (a \cdot c) \):
\[
\log_b (a \cdot c) = \log_b (b^x \cdot b^y)
\]
By the properties of exponents, we can combine \( b^x \) and \( b^y \):
\[
b^x \cdot b^y = b^{x+y}
\]
Applying the logarithm:
\[
\log_b (b^{x+y}) = x + y
\]
Now substituting \( x = \log_b a \) and \( y = \log_b c \):
\[
\log_b (a \cdot c) = \log_b a + \log_b c
\]
Thus, the product rule is proven.
### Quotient Rule of Logarithms
**Statement**:
If \( a > 0 \) and \( b > 0 \), then:
\[
\log_b \left( \frac{a}{c} \right) = \log_b a - \log_b c
\]
**Proof**:
Using the definition of logarithms:
\[
\log_b a = x \implies b^x = a
\]
\[
\log_b c = y \implies b^y = c
\]
Now we apply this to find \( \log_b \left( \frac{a}{c} \right) \):
\[
\log_b \left( \frac{a}{c} \right) = \log_b \left( \frac{b^x}{b^y} \right)
\]
Using the properties of exponents:
\[
\frac{b^x}{b^y} = b^{x-y}
\]
Applying the logarithm:
\[
\log_b \left( b^{x-y} \right) = x - y
\]
Now substituting \( x = \log_b a \) and \( y = \log_b c \):
\[
\log_b \left( \frac{a}{c} \right) = \log_b a - \log_b c
\]
Thus, the quotient rule is proven.
### Comparison of Product and Quotient Rules
- **Nature of Operations**: The product rule involves addition, resulting from the multiplicative combination of two positive quantities, while the quotient rule involves subtraction, resulting from the division of two positive quantities.
- **Application Context**: The product rule allows us to break down complex multiplicative expressions involving logarithms, while the quotient rule helps simplify expressions that involve division.
- **Formulation**:
- The product rule states that the logarithm of a product equals the sum of the logarithms:
\[
\log_b (a \cdot c) = \log_b a + \log_b c
\]
- The quotient rule states that the logarithm of a quotient equals the difference of the logarithms:
\[
\log_b \left( \frac{a}{c} \right) = \log_b a - \log_b c
\]
Both rules are critical for simplifying logarithmic expressions and finding solutions to equations involving logarithms.
### Product Rule of Logarithms
**Statement**:
If \( a > 0 \) and \( b > 0 \), then:
\[
\log_b (a \cdot c) = \log_b a + \log_b c
\]
**Proof**:
According to the definition of a logarithm, we have:
\[
\log_b a = x \implies b^x = a
\]
\[
\log_b c = y \implies b^y = c
\]
Now, we want to find \( \log_b (a \cdot c) \):
\[
\log_b (a \cdot c) = \log_b (b^x \cdot b^y)
\]
By the properties of exponents, we can combine \( b^x \) and \( b^y \):
\[
b^x \cdot b^y = b^{x+y}
\]
Applying the logarithm:
\[
\log_b (b^{x+y}) = x + y
\]
Now substituting \( x = \log_b a \) and \( y = \log_b c \):
\[
\log_b (a \cdot c) = \log_b a + \log_b c
\]
Thus, the product rule is proven.
### Quotient Rule of Logarithms
**Statement**:
If \( a > 0 \) and \( b > 0 \), then:
\[
\log_b \left( \frac{a}{c} \right) = \log_b a - \log_b c
\]
**Proof**:
Using the definition of logarithms:
\[
\log_b a = x \implies b^x = a
\]
\[
\log_b c = y \implies b^y = c
\]
Now we apply this to find \( \log_b \left( \frac{a}{c} \right) \):
\[
\log_b \left( \frac{a}{c} \right) = \log_b \left( \frac{b^x}{b^y} \right)
\]
Using the properties of exponents:
\[
\frac{b^x}{b^y} = b^{x-y}
\]
Applying the logarithm:
\[
\log_b \left( b^{x-y} \right) = x - y
\]
Now substituting \( x = \log_b a \) and \( y = \log_b c \):
\[
\log_b \left( \frac{a}{c} \right) = \log_b a - \log_b c
\]
Thus, the quotient rule is proven.
### Comparison of Product and Quotient Rules
- **Nature of Operations**: The product rule involves addition, resulting from the multiplicative combination of two positive quantities, while the quotient rule involves subtraction, resulting from the division of two positive quantities.
- **Application Context**: The product rule allows us to break down complex multiplicative expressions involving logarithms, while the quotient rule helps simplify expressions that involve division.
- **Formulation**:
- The product rule states that the logarithm of a product equals the sum of the logarithms:
\[
\log_b (a \cdot c) = \log_b a + \log_b c
\]
- The quotient rule states that the logarithm of a quotient equals the difference of the logarithms:
\[
\log_b \left( \frac{a}{c} \right) = \log_b a - \log_b c
\]
Both rules are critical for simplifying logarithmic expressions and finding solutions to equations involving logarithms.