Problem 42 - Precalculus

Question

Evaluate the expression. \(\ln \left(42^{6 \log (1)}\right)\)

Step-by-Step Solution

Answer

The expression evaluates to 0.

Step 1: Simplify Inside the Logarithm

Start by noticing the term inside the logarithm: \(6 \log(1)\). We can simplify this because we know that the logarithm of 1, in any base, is 0. Thus, \(\log(1) = 0\). Consequently, the entire expression becomes \(6 \times 0 = 0\).

Step 2: Evaluate the Exponent

Substitute the result from Step 1 back into the expression: \(42^{6 \times 0} = 42^0\). According to exponentiation rules, any non-zero number raised to the power of 0 is 1. Thus, \(42^0 = 1\).

Step 3: Evaluate the Natural Logarithm

Now that the expression inside the natural logarithm is simplified to 1, evaluate \(\ln(1)\). The natural logarithm of 1 is always 0. Therefore, \(\ln(1) = 0\).

Key Concepts

Logarithm Properties

Understanding logarithms is crucial for simplifying complex expressions involving these mathematical operations. Let's delve into a few essential properties, focusing on the natural logarithm, denoted as \(\ln\).

• The logarithm of 1, in any base, is 0. This is because any number raised to the power of zero equals 1. Consequently, \(\log_b(1) = 0\) for any base \(b\).
• Logarithms convert multiplication into addition: \(\log_b(xy) = \log_b(x) + \log_b(y)\). This property is advantageous for simplifying expressions.
• Another vital property is that logarithms turn exponentiation into multiplication: \(\log_b(x^y) = y \times \log_b(x)\).

Combining these properties can simplify even the most daunting logarithmic expressions. By recognizing these rules, handling calculations like the one in the original exercise becomes much easier.

Exponentiation Rules

Exponentiation might look intimidating at first, but knowing a few simple rules can make these operations much more approachable.One fundamental rule is that any non-zero number raised to the power of zero is always 1. This is expressed symbolically as \(a^0 = 1\) for any \(a eq 0\). Another important exponentiation rule states that when you multiply like bases, you can add the exponents: \(a^m \times a^n = a^{m+n}\).Moreover, raising a power to another power means you multiply the exponents: \((a^m)^n = a^{m \times n}\). Similarly, since multiplication is repetitive addition, exponentiation can be seen as repetitive multiplication.Understanding these rules helps swiftly simplify expressions involving powers, just as we utilized in evaluating the expression \(42^0\), leading to 1. This groundwork of understanding paves the way for resolving more complex operations with confidence.

Simplifying Expressions

Simplifying expressions is all about reducing them to their most basic form without changing their value, making them easier to work with.Start with simplifying any numbers or operations inside parentheses first. In the expression \(42^{6 \log(1)}\), we first noticed that \(\log(1) = 0\). This turned the exponent to zero, which simplified the expression exponentially!Next, apply algebraic properties: once you have simplified within parentheses, look at the exponentiation rules or logarithm properties that apply. For instance, in this exercise, applying the rule \(b^0 = 1\) led us to the simplified result.Finally, when everything else has been simplified, evaluate any outer functions or operations. With \(\ln(42^0)\) simplified to \(\ln(1)\), we applied our understanding that \(\ln(1) = 0\), leading us to the final, most simplified result.The goal is to recognize the various properties and rules and apply them step by step, leading to clarity and correct solutions.