Problem 25 - Contemporary Precalculus

Question

Evaluate the given expression without using a calculator. $$\log 10^{\sqrt{43}}$$

Step-by-Step Solution

Answer

Answer: \(\sqrt{43}\)

Step 1: Identify the logarithm properties

In this step, we will identify the logarithm properties that will help us evaluate the expression. One such property is the power rule which states that for any base \(b\), exponent \(n\) and a number \(x\): $$\log_b(x^n)=n\log_b(x)$$

Step 2: Apply the power rule

Applying the power rule to our given expression, we get: $$\log 10^{\sqrt{43}} = \sqrt{43}(\log 10)$$

Step 3: Simplify the expression

In this step, we will simplify the expression. We know that \(\log 10\) is equal to 1, because logarithm of a number with the same base is always equal to 1. Therefore, our expression can be simplified as: $$\sqrt{43}(\log 10) = \sqrt{43}(1)= \sqrt{43}$$

Step 4: State the final answer

Upon simplifying the given expression, we have the final answer: $$\log 10^{\sqrt{43}} = \sqrt{43}$$

Key Concepts

Power Rule

The Power Rule is one of the fundamental principles used when dealing with logarithmic functions. It helps us simplify expressions significantly. The rule states that the logarithm of a power can be rewritten by multiplying the exponent with the logarithm of the base. In mathematical terms, the Power Rule is expressed as follows:\[\log_b(x^n) = n \log_b(x)\]This means if you have a logarithm where a number is raised to an exponent, simply take the exponent in front of the logarithm. In the exercise, we use this rule to simplify \(\log 10^{\sqrt{43}}\) by transforming it into \(\sqrt{43}(\log 10)\). As you see, the exponent \(\sqrt{43}\) moves out in front as a multiplier. This rule greatly reduces the complexity of expressions involving large exponents or roots.

Logarithm Properties

Logarithm properties are essential tools in mathematics to simplify complex expressions. These properties allow us to manipulate and evaluate expressions without needing a calculator. Here are a few key properties you should be familiar with:

• Product Rule: \(\log_b(MN) = \log_b(M) + \log_b(N)\)
• Quotient Rule: \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\)
• Power Rule: \(\log_b(x^n) = n \log_b(x)\)
• Change of Base Formula: \(\log_b(x) = \frac{\log_a(x)}{\log_a(b)}\)

Using these properties can significantly ease the process of evaluating logarithmic expressions. In the problem, the Power Rule was particularly useful to transform and simplify the expression. Recognizing when and how to apply these properties is key to efficiently solving logarithmic equations.

Simplifying Expressions

Simplifying expressions, especially with logarithms, means rewriting an expression in its simplest or most efficient form. This often involves applying the properties of logarithms. In our problem \(\log 10^{\sqrt{43}}\), we first applied the Power Rule, resulting in \(\sqrt{43}(\log 10)\). To further simplify, we need to know some basics about logarithms:- \(\log 10\) is equal to 1 when the base of the log is 10. This is because \(10^1 = 10\). Therefore, any logarithm of a number where the number and the base are the same is 1.By substituting \(\log 10 = 1\) into our expression, it easily simplifies to \(\sqrt{43} \cdot 1 = \sqrt{43}\). This step highlights the power of understanding basic logarithmic values and ensures that you can recognize opportunities to simplify without a calculator. Mastering this skill is invaluable in mathematics and can make even complex-looking problems much more straightforward.