Problem 262 - Precalculus

Question

For the following exercises, condense to a single logarithm if possible. $$ \frac{1}{3} \ln (8) $$

Step-by-Step Solution

Answer

The expression condenses to \(\ln(2)\).

Step 1: Understand the Problem

We need to condense the expression \(\frac{1}{3} \ln (8)\) into a single logarithm. We will use logarithm properties to achieve this.

Step 2: Recall the Power Rule for Logarithms

The power rule for logarithms states: \(a \ln(b) = \ln(b^a)\). We will use this rule to simplify the expression \(\frac{1}{3} \ln (8)\).

Step 3: Apply the Power Rule

By applying the power rule \(a \ln(b) = \ln(b^a)\), substitute \(a = \frac{1}{3}\) and \(b = 8\). This gives us \(\ln(8^{\frac{1}{3}})\).

Step 4: Simplify the Expression

The expression \(8^{\frac{1}{3}}\) represents the cube root of 8, which is 2. Therefore, \(\ln(8^{\frac{1}{3}}) = \ln(2)\).

Key Concepts

Power Rule for Logarithms

The power rule for logarithms is a useful property that helps simplify expressions involving logarithms. Imagine that you have a logarithmic expression of the form \(a \ln(b)\). The power rule allows you to rewrite this as \(\ln(b^a)\). This transformation makes it easier to deal with the expression, especially if you need to combine or condense multiple logs.

Here's how it works:

• Take the coefficient or number in front of the logarithm (in this case, \(a\)).
• Use it as an exponent on the base of the logarithm (\(b\)).
• Now you have \(\ln(b^a)\), a single, condensed logarithm.

This property is particularly helpful when dealing with problems that require simplifying or combining logarithmic expressions, as seen in exercises like \(\frac{1}{3} \ln(8)\). By applying the power rule, we transform this into a more straightforward logarithmic form, making it easier to solve or simplify further.

Condensing to a Single Logarithm

Condensing an expression into a single logarithm involves using logarithmic properties to combine multiple logs into one. This process often uses the power, product, or quotient rules. In our specific example, we applied the power rule to achieve this.

When given a task to condense, follow these steps:

• Identify which logarithmic property to apply. Here, the power rule was used.
• Transform coefficients into exponents. For \(\frac{1}{3} \ln(8)\), the \(\frac{1}{3}\) becomes the exponent \(8^{\frac{1}{3}}\).
• Rewrite the expression as a single logarithm. It becomes \(\ln(8^{\frac{1}{3}})\).

This method simplifies the handling and understanding of logarithmic expressions, and makes further mathematical manipulation (like solving equations) less cumbersome. The goal is not just to simplify, but also to maintain the equivalence of the expression.

Simplified Logarithmic Expression

Simplifying a logarithmic expression means reducing it to its most basic form while retaining its original value. After applying any necessary properties, such as the power rule, you may still need to complete additional simplification.

For our example, we simplified \(\ln(8^{\frac{1}{3}})\) by evaluating the expression inside the logarithm:

• \(8^{\frac{1}{3}}\) is essentially the cube root of 8.
• The cube root of 8 is 2, since \(2^3 = 8\).

Hence, \(\ln(8^{\frac{1}{3}}) = \ln(2)\). By simplifying logarithmic expressions, calculations become more manageable, making it easier to understand and work with these mathematical functions in various applications.