Problem 33
Question
\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log \left(\frac{x^{3} y^{4}}{z^{6}}\right) $$
Step-by-Step Solution
Verified Answer
\(3\log(x) + 4\log(y) - 6\log(z)\)
1Step 1: Apply the Quotient Rule
According to the quotient rule of logarithms, which states that \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \), we can start expanding the expression. Apply this rule to the given logarithmic expression: \[ \log\left(\frac{x^3 y^4}{z^6}\right) = \log(x^3 y^4) - \log(z^6) \]
2Step 2: Apply the Product Rule
Now, applying the product rule of logarithms, which says \( \log(ab) = \log(a) + \log(b) \), we expand \( \log(x^3 y^4) \): \[ \log(x^3 y^4) = \log(x^3) + \log(y^4) \] Thus, the expression becomes: \[ \log(x^3) + \log(y^4) - \log(z^6) \]
3Step 3: Apply the Power Rule
Lastly, use the power rule of logarithms, which is \( \log(a^b) = b\log(a) \), to further expand each part of the expression: \[ \log(x^3) = 3\log(x), \; \log(y^4) = 4\log(y), \; \log(z^6) = 6\log(z) \] Substituting these into our expression, we get: \[ 3\log(x) + 4\log(y) - 6\log(z) \]
Key Concepts
Quotient Rule of LogarithmsProduct Rule of LogarithmsPower Rule of Logarithms
Quotient Rule of Logarithms
The Quotient Rule of Logarithms helps simplify complex expressions where two quantities are divided. According to this rule, the logarithm of a division can be split into the difference of two logarithms. Specifically, if you have an expression like \( \log\left(\frac{a}{b}\right) \), it can be rewritten as \( \log(a) - \log(b) \). This makes handling divisions much more straightforward.
In our original exercise, we applied the Quotient Rule to the expression \( \log\left(\frac{x^3 y^4}{z^6}\right) \), resulting in \( \log(x^3 y^4) - \log(z^6) \).
By breaking down a complex logarithmic function into simpler terms, it becomes easier to solve and understand.
In our original exercise, we applied the Quotient Rule to the expression \( \log\left(\frac{x^3 y^4}{z^6}\right) \), resulting in \( \log(x^3 y^4) - \log(z^6) \).
By breaking down a complex logarithmic function into simpler terms, it becomes easier to solve and understand.
Product Rule of Logarithms
When dealing with two values being multiplied within a logarithmic expression, the Product Rule of Logarithms offers a method to separate them. The rule states that the logarithm of a product can be expressed as the sum of the logarithms of the individual factors: \( \log(ab) = \log(a) + \log(b) \).
For our expression \( \log(x^3 y^4) \), using the Product Rule allows us to expand it into \( \log(x^3) + \log(y^4) \).
This simplification is beneficial, especially when the original expression is complex. It enables you to tackle each part individually and often reveals more about the behavior or properties of the function.
For our expression \( \log(x^3 y^4) \), using the Product Rule allows us to expand it into \( \log(x^3) + \log(y^4) \).
This simplification is beneficial, especially when the original expression is complex. It enables you to tackle each part individually and often reveals more about the behavior or properties of the function.
Power Rule of Logarithms
The Power Rule of Logarithms is particularly useful when dealing with exponents in logarithmic expressions. This rule states that a logarithm of a power can be simplified by taking the exponent in front of the logarithmic expression: \( \log(a^b) = b\log(a) \).
In our expanded expressions \( \log(x^3) \), \( \log(y^4) \), and \( \log(z^6) \), applying the Power Rule transforms them into \( 3\log(x) \), \( 4\log(y) \), and \( 6\log(z) \) respectively.
This method not only simplifies the expression but also makes it easier to interpret and manipulate in further mathematical operations or analyses.
In our expanded expressions \( \log(x^3) \), \( \log(y^4) \), and \( \log(z^6) \), applying the Power Rule transforms them into \( 3\log(x) \), \( 4\log(y) \), and \( 6\log(z) \) respectively.
This method not only simplifies the expression but also makes it easier to interpret and manipulate in further mathematical operations or analyses.
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