Problem 14 - College Algebra

Question

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

Step-by-Step Solution

Answer

The condensed single logarithm is \( \ln(2) \).

Step 1: Apply Power Rule of Logarithms

The power rule of logarithms states that you can bring a coefficient in front of the logarithm into the exponent of the argument. For this problem, we have \( \frac{1}{3} \ln(8) \). Apply the power rule: \( \frac{1}{3} \ln(8) = \ln(8^{1/3}) \).

Step 2: Simplify the Expression

Now, simplify the expression by evaluating the power of 8. The cube root of 8 is 2, i.e., \( 8^{1/3} = 2 \). Therefore, \( \ln(8^{1/3}) = \ln(2) \).

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a useful mathematical tool that simplifies expressions involving logarithms. It states that when you have a coefficient multiplied by a logarithm, you can move that coefficient to the exponent of the logarithm's argument.
For example, if you have an expression like \( c \cdot \ln(a) \), you can rewrite this using the power rule as \( \ln(a^c) \). This can be particularly helpful when condensing multiple logarithmic expressions or simplifying a single expression.

This rule is based on one of the properties of exponents, which allows a multiplication outside of the logarithm to shift into the exponent of the number inside the logarithm.
Applying this property can make complex logarithmic equations much easier to handle and solve. Using the power rule effectively requires understanding how exponents work and how they interact with logarithms.

Cube Root

A cube root is a number which when multiplied by itself three times, gives the original number. Let's consider the cube root of 8:

• The cube root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \).

Expressed in mathematical terms, the cube root of a number \( x \) is written as \( x^{1/3} \). In logarithmic contexts, recognizing cube roots can help with simplifying expressions,
especially when using rules like the power rule.

Understanding cube roots is important for simplifying logarithmic expressions as often these roots appear as exponents in various problems. By converting a number to its base roots, you can significantly simplify the calculation or transformation of the expression.
Challenge yourself to recognize patterns among cube roots for better efficiency in dealing with similar problems.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves the application of various logarithm rules in order to condense complex logarithmic expressions into more manageable forms.
In practice, this means using rules like the product, quotient, and power rules to rewrite log expressions.

The goal is to make the expression as simple as possible. In our example, by applying the power rule and then simplifying the resulting expression using the cube root, we condensed \( \frac{1}{3} \ln(8) \) into \( \ln(2) \).

• Apply the appropriate logarithmic rule such as power, product, or quotient rules as needed.
• Simplify using basic mathematical operations such as evaluating power or roots.

By mastering these techniques, students can solve even complicated logarithmic problems with ease. Always break down the expression into smaller parts, and apply these rules where applicable.
Understanding each step individually helps in mastering the overall simplification process.