Problem 30 - Algebra for College Students

Question

Evaluate each logarithmic expression. \(\log _{10} 10\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Logarithm Definition

The logarithm \(\log_{b}(x)\) refers to the exponent \(n\) to which the base \(b\) must be raised to yield the number \(x\). So, essentially, we are solving \(b^n = x\).

Step 2: Apply the Definition to the Given Expression

In the expression \(\log_{10} 10\), we are looking for the power \(n\) such that \(10^n = 10\).

Step 3: Solve the Equation

Since \(10^1 = 10\), we determine that \(n = 1\). Hence, \(\log_{10} 10 = 1\).

Key Concepts

Logarithm Definition

A logarithm answers the question: "To what power must we raise a base number to get another number?" In more formal terms, for a base \(b\) and a number \(x\), the logarithm \(\log_b(x)\) gives the power \(n\) such that \(b^n = x\). Understanding logarithms can help in solving equations where the variable is an exponent.

Here's a quick recap:

• The base \(b\) must be a positive number, not equal to 1.
• The number \(x\) for which we are finding the log must be positive.
• Logarithms are essentially the inverse of exponentiation.

Whenever you see a log, you're essentially asking the question of how many times you'd need to multiply the base by itself to reach the target number.

Base Exponentiation

Exponentiation is a mathematical operation involving a base and an exponent. In an expression like \(b^n\), \(b\) is the base, and \(n\) is the exponent, which indicates the number of times the base is multiplied by itself. For example, \(10^2\) would mean \(10\times10 = 100\).

Key points about exponentiation:

• If the exponent is 1, the base remains unchanged, i.e., \(b^1 = b\).
• If the exponent is 0, the result is always 1, provided the base is not zero, i.e., \(b^0 = 1\).
• Exponentiation is a way of expressing very large or small numbers conveniently.

Understanding how exponentiation works is essential when working with logarithms, as logs are simply the reverse of exponentiation.

Evaluating Logarithmic Expressions

Evaluating a logarithmic expression involves finding the power to which the base must be raised to get a specified number. For instance, when solving \(\log_{10}(10)\), we're seeking the exponent that makes \(10^n = 10\).

Steps to evaluate logarithmic expressions include:

• Identify both the base and the number inside the log.
• Rephrase the log expression into an equivalent exponential form.
• Solve for the exponent.

In the example \(\log_{10}(10)\), since \(10^1 = 10\), the exponent is 1, thus \(\log_{10}(10) = 1\). Recognizing this straightforward calculation can make other logarithmic evaluations much easier.