Problem 34 - Precalculus : Building Concepts and Connections

Question

Evaluate each expression without using a calculator. $$\log _{6} 6^{6 x}$$

Step-by-Step Solution

Answer

The value of \(\log _{6} 6^{6x}\) is \(6x\).

Step 1 - Rewrite the expression using a log identity

The logarithm identity \(\log _{b} b^{x} = x\) applies to the expression given. When the base of the log (\(b\)) and the base of the number inside the log (\(b^x\)) are the same, they cancel out, leaving only the exponent (\(x\)). Thus, the expression can be rewritten as: \(\log_{6} 6^{6x} = 6x\)

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

The value of \(\log _{6} 6^{6x}\) is \(6x\).

Key Concepts

Logarithm Identity

Logarithm identities are handy tools in simplifying complex problems involving logs. One essential identity is \(\log _{b} b^{x} = x\). This identity tells us that if a logarithm and its argument share the same base, they effectively "cancel" each other out, leaving just the exponent. This is because a logarithm is the inverse operation of exponentiation. When evaluating \(\log_{6} 6^{6x}\), you notice that both the base of the logarithm and the base of the exponent are 6. Thus, you can apply this identity directly to simplify the expression to \(6x\). This makes working with logarithmic expressions faster and less cumbersome, especially without a calculator. It's important to recognize when such identities apply as they make complex evaluations much simpler.

Exponentiation

Exponentiation is a mathematical operation involving numbers, where a base is raised to the power of an exponent. For instance, in the expression \(6^{6x}\), 6 is the base, and \(6x\) is the exponent. This means you multiply the base number (6) with itself a certain number of times dictated by the exponent.

• The base acts as a repeated factor.
• The exponent tells us how many times the base multiplies itself.

Understanding the relationship between exponents and logarithms is crucial. Since logarithms are the inverse of exponentiation, they serve as the means to "undo" the process, revealing the exponent when the base and result are known. So, when you see expressions like \(\log _{b} b^{x}\), you can swiftly recognize the simplification possible due to this inverse nature.

Mathematical Expressions

Mathematical expressions like \(\log _{6} 6^{6x}\) are combinations of numbers, variables, and operation symbols that together represent a value or a relationship. Evaluating expressions without a calculator often involves applying mathematical rules and identities to simplify the problem.

• Look for patterns: Identifying a log identity can be very beneficial.
• Break down the expression: Understanding each component, like the log base and the exponent, is key.
• Apply mathematical rules: Exponentiation and logarithms have specific rules that help simplify expressions.

Simplifying expressions isn't just about reaching an answer; it's about understanding how different mathematical operations interact. Recognizing identities and the inverse relationship between operations like logarithms and exponentiation helps in this process. Remember, mastering these components makes more complex mathematics easier to handle.