Problem 31 - Intermediate Algebra

Question

For Problems \(21-40\), evaluate each expression. $$ \log _{10} 0.1 $$

Step-by-Step Solution

Answer

The value of \(\log_{10} 0.1\) is -1.

Step 1: Understand the Logarithm Definition

The expression \(\log_{10} 0.1\) asks us to find the power to which the base 10 must be raised to produce 0.1. In general, \(\log_b a = c\) means \(b^c = a\). Here, the base \(b\) is 10, and \(a\) is 0.1.

Step 2: Set Up the Equation

Based on the logarithm definition, we set \(10^x = 0.1\), where \(x\) is the exponent we are trying to find. This will tell us what power of 10 equals 0.1.

Step 3: Rewrite 0.1 as a Power of 10

We know that 0.1 can be rewritten as \(\frac{1}{10}\) or \(10^{-1}\). So, the equation becomes \(10^x = 10^{-1}\).

Step 4: Solve the Exponent Equation

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

Key Concepts

Exponentiation

Exponentiation is a fundamental operation in mathematics. It involves raising a number, known as the base, to the power of an exponent. The expression \(b^n\) signifies that the base \(b\) is multiplied by itself \(n-1\) more times. So, for example, \(2^3 = 2 \times 2 \times 2 = 8\).
Exponentiation is a concept that conveys repeated multiplication. It is crucial in fields ranging from algebra to calculus. Understanding it allows us to comprehend powers of numbers, growth patterns, and even computing logarithms.
When dealing with exponentiation, there is a particular interest in specific bases like 10, which is commonly used in scientific notation and logarithms. Knowing how to manipulate and understand powers of 10 is especially useful in handling large or small values efficiently.

Base 10 Logarithm

The base 10 logarithm, often written as \(\log_{10}\) or just \(\log\) in many contexts, helps us find the power we need to raise 10 in order to get another number. It answers questions like: "How many tens do I multiply to get this number?" In other words, if we have \(\log_{10} a = c\), we mean that \(10^c = a\).
Base 10 logarithms are ubiquitous in science and engineering because they simplify dealing with very large or very small numbers. For instance:

• Sound intensity is often measured in logarithmic units called decibels.
• pH levels in chemistry are also logarithmic, indicating acidity or basicity.

In the exercise we solved, \(\log_{10} 0.1\) required us to rewrite 0.1 as \(10^{-1}\) because using exponent rules, \(0.1\) is essentially \(1/10\) which is \(10\) raised to the power of \(-1\). This demonstrates how logarithms work in reverse to exponentiation, solving for the exponent needed to achieve a certain number.

Solving Equations

Solving equations, particularly those involving logarithms or exponentials, involves finding the unknown variable that satisfies the equation. To solve an equation means to find all the values of the variable that make the equation true.
In our exercise, we started with \(10^x = 0.1\). By recognizing that 0.1 can be written as \(10^{-1}\), we changed the equation to \(10^x = 10^{-1}\). Since both sides have the same base, we equated the exponents, resulting in \(x = -1\).
This example illustrates a common method of solving equations - expressing both sides in terms of the same base.

• If \(b^x = b^y\), then \(x = y\).
• This property makes it simple to solve many exponential or logarithmic equations.

By converting the problem into a straightforward exponent equality, we leverage the property that identical bases allow for direct comparison of their exponents, simplifying the solution process.