Problem 37 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. Do not use a calculator. $$\log 10^{\sqrt{3}}$$

Step-by-Step Solution

Answer

The expression evaluates to \( \sqrt{3} \).

Step 1: Identify the Logarithm Rule

Recognize the property of logarithms used in this problem, which is the power rule. The power rule states that \( \log_b(a^c) = c \cdot \log_b(a) \).

Step 2: Apply the Power Rule

Using the power rule, rewrite the expression \( \log(10^{\sqrt{3}}) \) as \( \sqrt{3} \cdot \log(10) \).

Step 3: Evaluate the Known Logarithm

Since \( \log(10) \) with base 10 is 1 (because \(10^1 = 10\)), substitute this value into the expression. Thus, \( \sqrt{3} \cdot \log(10) = \sqrt{3} \cdot 1 \).

Step 4: Simplify the Expression

Multiply \( \sqrt{3} \) by 1 to simplify the expression. The result is \( \sqrt{3} \).

Key Concepts

Understanding the Power Rule

Logarithms have various rules that help simplify complex expressions. One of the most significant rules is the **power rule**. This rule states that when you have a logarithm of a power, it can be expressed by multiplying the exponent by the logarithm of the base.
For example, consider the expression \( \log_b(a^c) \). According to the power rule, this converts to:

• \( c \cdot \log_b(a) \)

This principle allows us to break down and manage logarithmic expressions that may initially seem difficult.
In problems like \( \log(10^{\sqrt{3}}) \), applying the power rule is pivotal. You separate the exponent (\( \sqrt{3} \)) from the logarithm, turning the expression into a straightforward product. Thus, the power rule simplifies the evaluation of logarithms, anchoring other calculations to more manageable arithmetic.

Exploring Base 10 Logarithms

Most students encounter logarithms with base 10, often referred to as **common logarithms**. These are ubiquitous in mathematics, especially in scientific calculations and real-world applications like measuring the pH in chemistry or calculating decibels in sound levels.
A base 10 logarithm, denoted as \( \log(\ldots) \), has the property that \( \log(10) = 1 \). This is because raising 10 to the power of 1 gives 10; hence \( 10^1 = 10 \).

• With this knowledge, any expression like \( \log(10^x) \) simplifies significantly because \( \log(10) \) will always result in 1.
• This simplification is crucial as seen in our step-by-step example: \( \log(10) = 1 \).

Recognizing this characteristic of base 10 logarithms aids in seamlessly evaluating expressions without requiring complex calculations or technologies.

Evaluating Expressions with Logarithms

When it comes to **evaluating expressions** involving logarithms, the process often involves using the rules associated with logarithmic identities.
To evaluate expressions such as \( \log(10^{\sqrt{3}}) \), follow these straightforward steps:

• First, identify the logarithmic property relevant to your expression, such as the power rule.
• Apply the rule to transform the expression into a simpler form. In this case, turn it into \( \sqrt{3} \cdot \log(10) \).
• Utilize known logarithm values (e.g., \( \log(10) = 1 \)) to continue simplifying.
• Substitute and compute to derive a clean result. Here, it becomes \( \sqrt{3} \cdot 1 = \sqrt{3} \).

Breaking down the operation into these steps ensures that you maintain accuracy while simplifying complex-looking logarithmic expressions. This approach is helpful for exams and dealing with logarithms in various mathematical contexts.