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Chapter 5: Problem 38

Express as a single logarithm and, if possible, simplify. $$\ln 54-\ln 6$$

Short Answer

Expert verified
\text{ln}(9)

Step by step solution

01

Apply the Difference Rule

Use the logarithm property \(\text{log}_{b}(M) - \text{log}_{b}(N) = \text{log}_{b}(\frac{M}{N})\) to combine the logarithms. Here, we have \(\text{ln}(54) - \text{ln}(6)\).
02

Simplify the Quotient

Rewrite the expression \(\text{ln}(\frac{54}{6})\) using the property from the previous step.
03

Simplify the Fraction

Divide \(\frac{54}{6} = 9\), so the expression becomes \(\text{ln}(9)\).

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Key Concepts

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

Difference Rule in Logarithms
To combine two logarithmic expressions with a subtraction operator, you can use the **Difference Rule**. The mathematical expression for this rule is: \[\log_{b}(M) - \log_{b}(N) = \log_{b}\left(\frac{M}{N}\right)\]
This means that when you subtract two logarithms with the same base, you can write them as a single logarithm of the ratio of their arguments.

In the given exercise, you start with \(\ln 54 - \ln 6\). Applying the difference rule, you change this to \( \ln \left(\frac{54}{6}\right) \).
Understanding this rule will help you simplify many logarithmic expressions easily.
Properties of Logarithms
Logarithms come with a set of handy properties that let us simplify and solve various expressions. Let's cover some important ones you might frequently encounter:
  • **Product Rule:** \(\log_{b} (MN) = \log_{b} M + \log_{b} N\)
  • **Quotient Rule:** \(\log_{b} \left(\frac{M}{N}\right) = \log_{b} M - \log_{b} N\)
  • **Power Rule:** \(\log_{b} (M^k) = k \log_{b} M \)

In our exercise, we used the **Quotient Rule** to combine \(\ln 54\) and \(\ln 6\) into a single logarithmic expression. Applying these properties can greatly simplify your work when handling logarithmic equations.
Simplifying Fractions within Logarithms
When working with logarithms that include fractions, you often need to simplify the fractions to make the logarithmic expression simpler.

In the exercise, after applying the **Difference Rule**, we got \(\ln \left(\frac{54}{6}\right)\).
The next step is to simplify the fraction \(\frac{54}{6}\). To do this, you divide 54 by 6, resulting in 9.

Now, the expression can be further simplified to \(\ln 9\).
Understanding how to simplify fractions helps manage more complex logarithmic forms and enhances your algebra skills.

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