Problem 27
Question
In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(\frac{x^{2} y}{z^{2}}\right) $$
Step-by-Step Solution
Verified Answer
The expanded form of the given logarithmic expression is \( 2\log_bx + \log_by - 2\log_bz \)
1Step 1: Apply the Logarithm of Quotient Rule
According to the logarithm of a quotient rule, \( \log_b(M/N) = \log_bM - \log_bN \). So let's apply this rule to the given expression. The fraction in the expression can be split into: \( \log_b\left(x^{2} y\right) - \log_b\left(z^{2}\right) \).
2Step 2: Apply the Logarithm of a Product Rule
The logarithm of a product rule states that \( \log_bMN = \log_bM + \log_bN \). Applying this rule to the \( \log_b\left(x^{2} y\right) \) results in: \( \log_b\left(x^{2}\right) + \log_b\left(y\right) - \log_b\left(z^{2}\right) \).
3Step 3: Apply the Power Rule of Logarithms
According to the power rule, \( \log_bM^n = n\log_bM \). So, applying this rule the expression can be further simplified to: \( 2\log_bx + \log_by - 2\log_bz \)
Key Concepts
Logarithm of Quotient RuleLogarithm of a Product RulePower Rule of Logarithms
Logarithm of Quotient Rule
Understanding the logarithm of a quotient rule is akin to unlocking a fundamental concept in algebra. This rule states, quite simply, that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator.
Let's unwrap this with an example. Given the expression \( \log_b\left(\frac{M}{N}\right) \) where \( b \) is the base, \( M \) is the numerator and \( N \) is the denominator, the quotient rule tells us that this can be simplified to: \[ \log_b(M) - \log_b(N) \.\] This is a powerful tool, as it helps to break down complex logarithmic expressions into more manageable pieces.
As seen in the exercise, when an expression like \( \log_b\left(\frac{x^2 y}{z^2}\right) \) is presented, applying the quotient rule is the first step, resulting in the separation of the terms within the logarithm: \( \log_b\left(x^2 y\right) - \log_b\left(z^2\right) \) . The next steps will involve further simplification using other logarithmic properties.
Let's unwrap this with an example. Given the expression \( \log_b\left(\frac{M}{N}\right) \) where \( b \) is the base, \( M \) is the numerator and \( N \) is the denominator, the quotient rule tells us that this can be simplified to: \[ \log_b(M) - \log_b(N) \.\] This is a powerful tool, as it helps to break down complex logarithmic expressions into more manageable pieces.
As seen in the exercise, when an expression like \( \log_b\left(\frac{x^2 y}{z^2}\right) \) is presented, applying the quotient rule is the first step, resulting in the separation of the terms within the logarithm: \( \log_b\left(x^2 y\right) - \log_b\left(z^2\right) \) . The next steps will involve further simplification using other logarithmic properties.
Logarithm of a Product Rule
The logarithm of a product rule is another key instrument in the toolbox of algebra. It teaches us how to handle the logarithm of a multiplicative combination of variables. The rule is intuitive: the logarithm of a product is equal to the sum of the logarithms of the individual factors.
The formula for this rule is \( \log_b(MN) = \log_b(M) + \log_b(N) \) where \( M \) and \( N \) are the factors being multiplied. For instance, if we have a product like \( x^2y \) within a logarithmic function, we can apply the rule to split it into two separate logs, yielding \( \log_b\left(x^2\right) + \log_b(y) \) .
In the context of our exercise, the expression \( \log_b\left(x^2 y\right) \) quickly becomes \( \log_b\left(x^2\right) + \log_b(y) \) after this rule is applied. As we continue to solve, each of these logs can then be individually addressed, often leading to simpler forms or even final values for the variables involved.
The formula for this rule is \( \log_b(MN) = \log_b(M) + \log_b(N) \) where \( M \) and \( N \) are the factors being multiplied. For instance, if we have a product like \( x^2y \) within a logarithmic function, we can apply the rule to split it into two separate logs, yielding \( \log_b\left(x^2\right) + \log_b(y) \) .
In the context of our exercise, the expression \( \log_b\left(x^2 y\right) \) quickly becomes \( \log_b\left(x^2\right) + \log_b(y) \) after this rule is applied. As we continue to solve, each of these logs can then be individually addressed, often leading to simpler forms or even final values for the variables involved.
Power Rule of Logarithms
The power rule of logarithms is pivotal in simplifying logarithmic expressions with exponents. Put simply, the power rule allows us to move the exponent in a logarithmic argument out in front as a multiplier.
This rule is expressed mathematically as \( \log_b(M^n) = n\log_b(M) \) where \( n \) is the exponent. This means that if you have an expression like \( \log_b\left(x^2\right) \) you can rewrite it as \( 2\log_b(x) \), transforming an exponent into a coefficient.
Going through our exercise, the power rule significantly reduces the complexity of our initial logarithm. We can take the power of each term in the logarithmic expression and bring it out in front, thus \( \log_b\left(x^2\right) \) becomes \( 2\log_b(x) \) and similarly, \( \log_b\left(z^2\right) \) becomes \( 2\log_b(z) \), allowing us to see the logarithm of the original variables without their exponents clouding the equation.
This rule is expressed mathematically as \( \log_b(M^n) = n\log_b(M) \) where \( n \) is the exponent. This means that if you have an expression like \( \log_b\left(x^2\right) \) you can rewrite it as \( 2\log_b(x) \), transforming an exponent into a coefficient.
Going through our exercise, the power rule significantly reduces the complexity of our initial logarithm. We can take the power of each term in the logarithmic expression and bring it out in front, thus \( \log_b\left(x^2\right) \) becomes \( 2\log_b(x) \) and similarly, \( \log_b\left(z^2\right) \) becomes \( 2\log_b(z) \), allowing us to see the logarithm of the original variables without their exponents clouding the equation.
Other exercises in this chapter
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