Problem 30 - Algebra 2 and Trigonometry

Question

In \(27-56,\) evaluate each logarithmic expression. Show all work. $$ 4 \log _{6} 216 $$

Step-by-Step Solution

Answer

The value of \(4 \log_{6} 216\) is 12.

Step 1: Simplify the Expression

The given expression is \(4 \log_{6} 216\). Begin by understanding that this expression involves multiplying 4 by \(\log_{6} 216\). This can be rewritten by using the logarithm power rule: \(a \log_b c = \log_b (c^a)\). Thus, \(4 \log_{6} 216 = \log_{6} (216^4)\).

Step 2: Evaluate the Logarithm

To find \(\log_{6} (216^4)\), first evaluate \(216^4\). Then, express 216 as a power of 6. Start by factoring 216: \(216 = 6^3\). Then calculate \((6^3)^4 = 6^{12}\). Therefore, \(\log_{6} (6^{12})\).

Step 3: Apply the Logarithm Identity

Use the logarithm identity \(\log_b (b^c) = c\). Substituting in our expression, \(\log_{6} (6^{12}) = 12\). Thus, the value of \(4 \log_{6} 216\) is 12.

Key Concepts

Logarithm Power Rule

The logarithm power rule is a handy mathematical tool for simplifying expressions. It states that if you have a logarithm in the form of \(a \log_b c\), you can transform it into \(\log_b (c^a)\). This rule allows you to "power up" the argument of the logarithm.

For example, consider the expression \(4 \log_{6} 216\). By applying the logarithm power rule, we convert this to \(\log_{6} (216^4)\).

• This change reduces the complexity of calculations by raising \(216\) to the power of 4.
• Such transformations often simplify your journey to the final solution because it allows further manipulation and easier evaluation.

This rule is incredibly useful when you deal with expressions involving coefficients in front of logarithms, guiding you directly to simplify and solve them efficiently.

Logarithm Evaluation

Evaluating logarithms is a process of finding the exponent needed to raise a base to get a given number. In our example, \(216\) becomes pivotal when expressing it as a power of \(6\). We start by identifying that 216 can be expressed as \(6^3\), meaning that \(6^3 = 216\).

Next, calculate \((6^3)^4 = 6^{12}\). Thus, the expression \(\log_{6} (216^4)\) simplifies to \(\log_{6} (6^{12})\), which we will evaluate next.

• This strategy of expressing the number as a power of the base streamlines the process of solving logarithms.
• Breaking down numbers in this way helps in developing a clearer understanding of the relationships within logarithmic expressions.

Through such evaluations, we better grasp the underlying mechanics that simplify complex logarithmic expressions.

Logarithmic Identities

Logarithmic identities allow mathematicians to simplify complex expressions. One of the most powerful identities is \(\log_b (b^c) = c\). This indicates that a logarithm of a base raised to some power directly simplifies to that power.

In the example problem, we reduced \(\log_{6} (6^{12})\) using this identity.

• Substituting directly provides \(\log_{6} (6^{12}) = 12\), as the base \(6\) matches the base used in the expression.
• Forging a direct path to the solution, logarithmic identities like this eliminate unnecessary steps and focus on direct results.

Overall, mastering these identities empowers you to conquer logarithmic assessments efficiently, revealing underlying simplicity in seemingly complex calculations.