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Chapter 5: Problem 30

Write the expression as a logarithm of a single quantity. $$ 3 \ln x+2 \ln y-4 \ln z $$

Short Answer

Expert verified
The simplified expression is \(\ln\left(\frac{x^3y^2}{z^4}\right)\)

Step by step solution

01

Applying Logarithm Power Rule

Firstly, apply the logarithm power rule, which states that \(\log_a(m^n) = n\log_a(m)\), to each term to simplify the expression. This turns the expression into \(\ln(x^3) + \ln(y^2) - \ln(z^4)\)
02

Applying the Logarithm Product Rule

Next, apply the logarithm product rule which states that \(\log_a(m \cdot n) = \log_a(m) + \log_a(n)\). Apply it to the first two terms: \(\ln(x^3) + \ln(y^2)\) which simplifies to \(\ln(x^3y^2)\)
03

Applying the Logarithm Division Rule

Then, apply the logarithm division rule which states that \(\log_a\left(\frac{m}{n}\right) = \log_a(m) - \log_a(n)\). Apply the rule to the whole expression: \(\ln(x^3y^2) - \ln(z^4)\), which simplifies to \(\ln\left(\frac{x^3y^2}{z^4}\right)\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm Power Rule
The logarithm power rule is a handy tool that helps simplify logarithmic expressions. It tells us that we can bring the exponent in a logarithmic term to the front of the logarithm. Mathematically, this is expressed as:
  • \( \log_a(m^n) = n \log_a(m) \)
So, if we have a term like \( 3 \ln x \), we can apply the power rule to rewrite it as \( \ln(x^3) \). Similarly, \( 2 \ln y \) becomes \( \ln(y^2) \), and \( 4 \ln z \) becomes \( \ln(z^4) \).
This step is crucial because it transforms each logarithmic term into a more compact form, setting the stage for further simplification.
Logarithm Product Rule
The logarithm product rule is another essential identity and comes in handy when combining logarithms. It states that the logarithm of a product is the sum of the logarithms of the individual factors. In formula form:
  • \( \log_a(m \cdot n) = \log_a(m) + \log_a(n) \)
Using this rule, two logarithms that are added together, like \( \ln(x^3) + \ln(y^2) \), can be combined into a single logarithm: \( \ln(x^3 y^2) \).
This simplification combines terms into one logarithmic expression, making the entire expression easier to manage and further manipulate. It's akin to collecting like terms in algebra.
Logarithm Division Rule
The logarithm division rule provides a way to combine logarithms of two terms being subtracted. This rule states that the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator:
  • \( \log_a\left(\frac{m}{n}\right) = \log_a(m) - \log_a(n) \)
In the final step of our simplification, we use this rule to combine the expression \( \ln(x^3 y^2) - \ln(z^4) \). This is rewritten as a single logarithm: \( \ln\left(\frac{x^3 y^2}{z^4}\right) \).
This transformation not only merges multiple logarithmic expressions into one but also provides a clearer and simpler form to interpret or evaluate. Once you have mastered these three fundamental logarithmic identities, transforming complex logarithmic expressions becomes much more straightforward.

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