Problem 33
Question
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
The quotient rule for logarithms states that \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\). Apply this to the given function: \(\log\left(\frac{x^{3} y^{4}}{z^{6}}\right)\). This gives us: \(\log(x^{3}y^{4}) - \log(z^{6})\).
2Step 2: Apply the Product Rule
The product rule for logarithms states that \(\log(ab) = \log(a) + \log(b)\). Apply this to the expression \(\log(x^{3}y^{4})\). This gives us \(\log(x^{3}) + \log(y^{4})\).
3Step 3: Apply the Power Rule
Use the power rule for logarithms, which states that \(\log(a^{b}) = b\cdot\log(a)\). Apply this to each term: \(\log(x^{3}) = 3\cdot\log(x)\) and \(\log(y^{4}) = 4\cdot\log(y)\). Also \(\log(z^{6}) = 6\cdot\log(z)\).
4Step 4: Combine the Results
Combine all terms from the previous steps: \(\log(x^{3} y^{4}) - \log(z^{6}) = 3\cdot\log(x) + 4\cdot\log(y) - 6\cdot\log(z)\). This is the expanded form of the logarithmic expression.
Key Concepts
Laws of LogarithmsQuotient RuleProduct RulePower Rule
Laws of Logarithms
Logarithms have several important properties that help simplify complex expressions. Among these, there are three fundamental laws: the product rule, quotient rule, and power rule. These rules transform logarithmic expressions into simpler or more manageable forms. They are essential tools for solving logarithmic equations and for mathematical problem-solving.
Understanding the laws of logarithms enables you to expand or condense logarithmic expressions effectively. The idea is to use these rules systematically to break down an expression into smaller, more digestible parts or vice versa. Overall, the laws of logarithms make it easier for you to manipulate and solve problems that involve exponential relationships.
Understanding the laws of logarithms enables you to expand or condense logarithmic expressions effectively. The idea is to use these rules systematically to break down an expression into smaller, more digestible parts or vice versa. Overall, the laws of logarithms make it easier for you to manipulate and solve problems that involve exponential relationships.
Quotient Rule
The quotient rule for logarithms states that if you have a logarithm that divides one expression by another, you can rewrite it as the difference of the logarithms of these expressions.
Mathematically, it is expressed as: \[ \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \]
This rule is invaluable when you need to break down divisive logarithmic expressions into simpler parts that can be individually analyzed or manipulated.
In our original exercise, we applied the quotient rule to the given expression \( \log\left(\frac{x^{3} y^{4}}{z^{6}}\right) \), transforming it into \( \log(x^{3}y^{4}) - \log(z^{6}) \). By doing this, we split the expression into two distinct parts, each of which could then be further simplified using other logarithm rules.
Mathematically, it is expressed as: \[ \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \]
This rule is invaluable when you need to break down divisive logarithmic expressions into simpler parts that can be individually analyzed or manipulated.
In our original exercise, we applied the quotient rule to the given expression \( \log\left(\frac{x^{3} y^{4}}{z^{6}}\right) \), transforming it into \( \log(x^{3}y^{4}) - \log(z^{6}) \). By doing this, we split the expression into two distinct parts, each of which could then be further simplified using other logarithm rules.
Product Rule
Whenever you have a logarithm of a product, the product rule allows you to separate it into a sum of logarithms. This is expressed as:
\[ \log(ab) = \log(a) + \log(b) \]
The product rule simplifies expressions involving multiplication within logarithms. By breaking down a product into individual log terms, you can address each part separately, as shown in our example.
In step 2 of the solution, we applied the product rule to \( \log(x^{3}y^{4}) \), converting it into \( \log(x^{3}) + \log(y^{4}) \). This dissection allows for further simplification using the power rule, showing how rules can be used in combination to simplify complex expressions efficiently.
\[ \log(ab) = \log(a) + \log(b) \]
The product rule simplifies expressions involving multiplication within logarithms. By breaking down a product into individual log terms, you can address each part separately, as shown in our example.
In step 2 of the solution, we applied the product rule to \( \log(x^{3}y^{4}) \), converting it into \( \log(x^{3}) + \log(y^{4}) \). This dissection allows for further simplification using the power rule, showing how rules can be used in combination to simplify complex expressions efficiently.
Power Rule
The power rule states that any logarithmic expression involving an exponent can be rewritten by bringing the exponent in front of the logarithm as a multiplier. It is expressed as:
\[ \log(a^{b}) = b\cdot\log(a) \]
This rule is particularly helpful when dealing with expressions where variables are raised to a power.
In our problem, the power rule was applied to each part of the expression resulting from earlier steps. We transformed \( \log(x^{3}) \) into \( 3\cdot\log(x) \), \( \log(y^{4}) \) into \( 4\cdot\log(y) \), and \( \log(z^{6}) \) into \( 6\cdot\log(z) \).
Bringing down the exponents in this way makes it much easier to work with and compare logarithmic expressions. Finally, the results were combined to get the expanded expression of the original problem.
\[ \log(a^{b}) = b\cdot\log(a) \]
This rule is particularly helpful when dealing with expressions where variables are raised to a power.
In our problem, the power rule was applied to each part of the expression resulting from earlier steps. We transformed \( \log(x^{3}) \) into \( 3\cdot\log(x) \), \( \log(y^{4}) \) into \( 4\cdot\log(y) \), and \( \log(z^{6}) \) into \( 6\cdot\log(z) \).
Bringing down the exponents in this way makes it much easier to work with and compare logarithmic expressions. Finally, the results were combined to get the expanded expression of the original problem.
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