Problem 68 - Algebra A Combined Function

Question

Evaluate each expression. $$ \log _{10} \frac{1}{10} $$

Step-by-Step Solution

Answer

The value of \( \log_{10} \frac{1}{10} \) is \(-1\).

Step 1: Understanding the Problem

We need to evaluate the logarithm of the fraction \( \frac{1}{10} \) with base 10, which is expressed as \( \log_{10} \frac{1}{10} \). The goal is to find what power 10 must be raised to in order to yield \( \frac{1}{10} \).

Step 2: Recall the Definition of Logarithms

The logarithmic expression \( \log_{b}(a) = x \) is equivalent to the exponential expression \( b^x = a \). In this case, we'll convert the logarithm \( \log_{10} \frac{1}{10} \) into its equivalent exponential form.

Step 3: Convert to Exponential Form

Convert the logarithmic expression \( \log_{10} \frac{1}{10} \) to exponential form: \( 10^x = \frac{1}{10} \). We need to determine the value of \( x \) that makes this equation true.

Step 4: Express the Fraction as a Power of 10

Recognize that \( \frac{1}{10} \) can be rewritten as \( 10^{-1} \). Therefore, the equation becomes \( 10^x = 10^{-1} \).

Step 5: Solve for x

Since the bases are the same (both are 10), we can equate the exponents. This gives us the equation \( x = -1 \).

Step 6: Verify the Solution

Check the solution by substituting back into the exponential equation: If \( x = -1 \), then \( 10^{-1} = \frac{1}{10} \). This confirms our solution is correct.

Key Concepts

Properties of Logarithms

Logarithms are powerful tools in mathematics that help simplify complex multiplications and divisions into basic additions and subtractions. They are particularly useful in solving equations where the unknown is in the exponent. To better understand logarithms, it’s key to grasp their main properties, which can simplify mathematical tasks:

Basic Properties:

• Product Property: The logarithm of a product is the sum of the logarithms: \( \log_b(mn) = \log_b(m) + \log_b(n) \).
• Quotient Property: The logarithm of a quotient is the difference of the logarithms: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \).
• Power Property: The logarithm of a power is the exponent times the logarithm: \( \log_b(m^n) = n \cdot \log_b(m) \).

Understanding these properties allows mathematicians and students to rewrite and solve seemingly difficult equations more manageably. For example, in our original problem \( \log_{10} \frac{1}{10} \), knowing the quotient property directly leads to a simpler way to express \( \log_{10} 1 - \log_{10} 10 \), which then clarifies why the result is -1.

Exponential Expressions

Exponential expressions are all about numbers involving exponents, where a base number is raised to a certain power. They are essential in working with logarithms because logarithms are essentially the reverse of exponentiation. In understanding how to solve a logarithmic expression, one must think in terms of its corresponding exponential form.

Converting from Logarithmic to Exponential:

• Given a logarithmic expression \( \log_b(a) = x \), this is equivalent to the exponential form \( b^x = a \).
• This helps transform what might seem like an abstract problem into a more intuitive one. For instance, the conversion of our problem \( \log_{10} \frac{1}{10} \) turns it into the exponential equation \( 10^x = \frac{1}{10} \).

This understanding transforms the problem of determining a logarithm into solving for the exponent in an exponential equation. It highlights how closely linked logarithms and exponents are, and why knowing how to convert between the two forms is crucial.

Mathematical Problem-Solving

Successful mathematical problem-solving requires a blend of strategy, understanding, and practice, especially when dealing with logarithmic or exponential expressions. Here are some key strategies to tackle these problems:

Step-by-Step Approach:

• Understand the Problem: Know what is being asked and identify what you’re trying to find. Decoding the language of mathematics is the first step.
• Recall Definitions and Properties: Remember, laws and identities like those related to logarithms or exponents are tools to use in solution paths.
• Convert and Rewrite: If necessary, convert logarithmic forms to exponential ones or vice versa, to see the problem from different angles.
• Solve and Verify: Solve the rewritten expressions carefully, keeping track of units and conversions. Always verify your answer with the original question to ensure accuracy.

By adhering to a methodical approach, students can demystify complex expressions and find satisfaction in solving problems accurately and confidently. This approach encourages thinking critically and applying known principles creatively.