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Magazine Chapter 1: Problem 20
Write the given quantity in terms of \(\log x, \log y,\) and \(\log z\). \(\log \frac{x y^{2}}{z^{3}}\)
Short Answer
Expert verified
\( \log x + 2 \log y - 3 \log z \)
Step by step solution
01
Understand the Problem
We need to express the given expression \( \log \frac{x y^{2}}{z^{3}} \) in terms of \( \log x \), \( \log y \), and \( \log z \). This involves using the properties of logarithms to break down the expression.
02
Apply the Division Rule
The division rule of logarithms states that \( \log \frac{A}{B} = \log A - \log B \). Applying this to our expression, we get: \( \log \frac{x y^{2}}{z^{3}} = \log(x y^{2}) - \log(z^{3}) \).
03
Apply the Multiplication Rule
The multiplication rule of logarithms states that \( \log(AB) = \log A + \log B \). Applying this to \( \log(x y^{2}) \), we have: \( \log(x y^{2}) = \log x + \log y^{2} \).
04
Apply the Power Rule
The power rule of logarithms states that \( \log(A^n) = n \cdot \log A \). We apply this rule to both \( \log y^{2} \) and \( \log z^{3} \): - \( \log y^{2} = 2 \cdot \log y \)- \( \log z^{3} = 3 \cdot \log z \)
05
Combine Like Terms
Using the results from previous steps, substitute back into \( \log \frac{x y^{2}}{z^{3}} = \log x + 2 \cdot \log y - 3 \cdot \log z \). This is the expression in terms of \( \log x \), \( \log y \), and \( \log z \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Properties of Logarithms
Logarithms come with a set of powerful properties that help simplify complex expressions. These properties are essential when dealing with logarithmic expressions, as they provide a handy way to manipulate and rearrange terms.
These are three main logarithmic properties:
These are three main logarithmic properties:
- Product (Multiplication) Rule: This rule states that the logarithm of a product is the sum of the logarithms of the factors. For example, \( \log(AB) = \log A + \log B \).
- Quotient (Division) Rule: This indicates that the logarithm of a quotient is the difference between the logarithm of the numerator and denominator: \( \log \frac{A}{B} = \log A - \log B \).
- Power Rule: The power rule tells us that the logarithm of an exponentiated term can be simplified by moving the exponent in front of the logarithm, like this: \( \log(A^n) = n \cdot \log A \).
Division Rule of Logarithms
The division rule, also known as the quotient rule, plays a significant role in breaking down complex logarithmic expressions. Specifically, it allows us to express the logarithm of a fraction.
According to this rule, if you have a logarithm of a quotient such as \( \log \frac{A}{B} \), you can rewrite it as the difference \( \log A - \log B \).
For instance, in our example, applying this rule to \( \log \frac{x y^{2}}{z^{3}} \) results in the expression \( \log(x y^{2}) - \log(z^{3}) \).
By separating the elements of the fraction, more intricate expressions become simpler to handle, making the division rule a valuable tool in mathematics.
According to this rule, if you have a logarithm of a quotient such as \( \log \frac{A}{B} \), you can rewrite it as the difference \( \log A - \log B \).
For instance, in our example, applying this rule to \( \log \frac{x y^{2}}{z^{3}} \) results in the expression \( \log(x y^{2}) - \log(z^{3}) \).
By separating the elements of the fraction, more intricate expressions become simpler to handle, making the division rule a valuable tool in mathematics.
Multiplication Rule of Logarithms
The multiplication rule of logarithms helps us simplify expressions that involve the product of multiple terms. It states that when you take the logarithm of a product, you can split it into a sum of separate logarithms.
The formal representation is: \( \log(AB) = \log A + \log B \).
In the context of our example, we've applied this rule to the expression \( \log(x y^{2}) \), which breaks down into \( \log x + \log y^{2} \).
This step is crucial as it further simplifies the expression, preparing it for the application of the power rule, which will separate out the exponents.
The formal representation is: \( \log(AB) = \log A + \log B \).
In the context of our example, we've applied this rule to the expression \( \log(x y^{2}) \), which breaks down into \( \log x + \log y^{2} \).
This step is crucial as it further simplifies the expression, preparing it for the application of the power rule, which will separate out the exponents.
Power Rule of Logarithms
The power rule of logarithms simplifies expressions where a term is raised to a power. This rule is useful in turning these types of expressions into a product involving the exponent.
The rule is mathematically expressed as \( \log(A^n) = n \cdot \log A \).
In our given problem, we have used the power rule for two terms: \( y^2 \) and \( z^3 \). Here, \( \log y^{2} \) becomes \( 2 \cdot \log y \) and \( \log z^{3} \) is rewritten as \( 3 \cdot \log z \).
By applying this rule, exponents are shifted from complicating the argument to simple multipliers, paving the way for easier manipulation of logarithmic expressions in algebraic transformations.
The rule is mathematically expressed as \( \log(A^n) = n \cdot \log A \).
In our given problem, we have used the power rule for two terms: \( y^2 \) and \( z^3 \). Here, \( \log y^{2} \) becomes \( 2 \cdot \log y \) and \( \log z^{3} \) is rewritten as \( 3 \cdot \log z \).
By applying this rule, exponents are shifted from complicating the argument to simple multipliers, paving the way for easier manipulation of logarithmic expressions in algebraic transformations.
