Problem 38 - A Graphical Approach to College Algebra

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: Understand the Problem

We are tasked with evaluating the expression \( \log 10^{\sqrt{3}} \). This is a logarithmic expression where the base of the logarithm is 10 (this is implied in the notation \( \log \)).

Step 2: Apply the Logarithmic Identity

Recall the identity for logarithms that states \( \log_b (b^x) = x \). This identity will help us simplify the given expression where the base \( b = 10 \). Hence, \( \log 10^{\sqrt{3}} \) can be simplified directly using this identity.

Step 3: Simplify Using the Identity

Using the identity \( \log_b (b^x) = x \), substitute \( b = 10 \) and \( x = \sqrt{3} \). Therefore, \( \log 10^{\sqrt{3}} \) simplifies to \( \sqrt{3} \).

Step 4: Verify the Result Mathematically

Check the steps logically: since \( \log 10^{\sqrt{3}} \) directly applies the identity \( \log_b (b^x) = x \), we confirm that \( x = \sqrt{3} \) is the accurate and valid solution.

Key Concepts

Logarithmic Identities

Logarithmic identities are powerful tools that make working with log expressions easier. One of the most useful identities is:

• \( \log_b (b^x) = x \)

This identity states that if the base of the logarithm and the base of the exponent are the same, the result is simply the exponent itself. This simplifies calculations dramatically.
In the given exercise, where you evaluate \( \log 10^{\sqrt{3}} \), applying this identity straightaway gives the answer \( \sqrt{3} \) since the base 10 of the logarithm matches the base of the exponent.
Learning these identities can speed up solving logarithmic problems without needing to compute complex calculations. Understanding the application of identities is crucial in improving problems solving skills involving logs.

Simplifying Expressions

Simplifying expressions involves reducing complex expressions into more manageable ones. For logarithms, using identities like \( \log_b (b^x) = x \) can turn a complex expression into a simple number.
Let's break down the simplification process:

• First, identify the base of the logarithm. Ensure it matches the base of the term inside the log.
• Apply the identity because it leads directly to a simpler expression that's easier to comprehend and solve.

In the exercise, you start with \( \log 10^{\sqrt{3}} \). Since its base (10) matches with the helps of logarithmic identity, you simplify it directly to \( \sqrt{3} \).
Use these strategies whenever simplifying logarithmic expressions. They are designed to enhance understanding and accuracy.

Base of Logarithms

The base of a logarithm is a key factor in understanding and simplifying logarithmic expressions. Logarithms automatically imply base 10 (common logarithm) unless specified differently such as \( \ln \) which denotes natural logarithms with base \( e \).
In our worked example, \( \log 10^{\sqrt{3}} \), the base is 10. Recognizing these bases quickly can guide you in applying identities accurately.
Consider these points:

• If there is no written base, assume it's 10.
• Matching bases in problem-solving eases application of identities like \( \log_b (b^x) = x \).

Understanding what the base implies and selecting create strategies based on it can make evaluating and simplifying logarithmic expressions effortless.