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Chapter 1: Problem 39

Express the given quantity as a single logarithm. $$ \ln 5+5 \ln 3 $$

Short Answer

Expert verified
\(\ln 1215\)

Step by step solution

01

Applying the Power Rule

First, we use the power rule of logarithms, which states that \( a \ln b = \ln(b^a) \). Apply this rule to \(5 \ln 3\) to get \(\ln(3^5)\). This simplifies to \(\ln 243\).
02

Combining Using the Product Rule

Next, use the product rule of logarithms, which states that \( \ln a + \ln b = \ln(ab) \). Apply this rule to combine \(\ln 5\) and \(\ln 243\) into one term: \(\ln(5 \times 243)\).
03

Simplifying the Expression

Calculate the product inside the logarithm: \(5 \times 243 = 1215\). Hence, the expression becomes \(\ln 1215\).

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Key Concepts

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

Power Rule in Logarithms
Logarithms often involve exponents and the power rule is a handy tool to simplify such expressions. The power rule states:
  • If you have a logarithmic expression like \( a \ln b \), it can be rewritten as \( \ln(b^a) \).
This means that you "move" the constant factor as an exponent of the base inside the logarithm. For example, in the expression \( 5 \ln 3 \), the number 5 is moved as an exponent of 3.
This changes the expression to \( \ln(3^5) \), which simplifies to \( \ln 243 \) because \( 3^5 = 243 \).
This rule is particularly useful when you need to combine logarithmic terms or make computations easier.
Product Rule in Logarithms
When you have two logarithms with the same base being added, you can use the product rule to combine them into a single logarithm. This rule states:
  • \( \ln a + \ln b = \ln(ab) \)
In our example, after using the power rule, we end up with \( \ln 5 \) and \( \ln 243 \).
According to the product rule, these can be combined into a single logarithm: \( \ln(5 \times 243) \).
This allows you to condense your expression, making it simpler and often easier to solve or further manipulate. Using this step reduces multiple terms to just one, which can be helpful in algebraic manipulations.
Simplifying Logarithmic Expressions
Simplifying logarithmic expressions involves applying rules of logarithms to rewrite them in a more compact or easier-to-use form.
After applying both the power and product rules in our example, you reach the point where you simplify the multiplication inside the logarithm:
  • Calculate \( 5 \times 243 \), which equals 1215.
Thus, the complex expression \( \ln 5 + 5 \ln 3 \) simplifies down to \( \ln 1215 \).
Simplified expressions like this are often more manageable and can provide clearer insights in further mathematical analyses or problem-solving scenarios.
Breaking down expressions into their simplest form can help you both in understanding problems better and in achieving solutions more efficiently.

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