Problem 32 - Intermediate Algebra

Question

For Problems \(21-40\), evaluate each expression. $$ \log _{10} 0.0001 $$

Step-by-Step Solution

Answer

\(\log_{10}(0.0001) = -4\)

Step 1: Recall the Logarithm Definition

The logarithm function tells us what power we need to raise a base to in order to obtain a certain number. In this case, the base is 10. So, we want to find the power of 10 that gives us 0.0001.

Step 2: Express 0.0001 as a Power of 10

Rewrite 0.0001 as a power of 10. We know that 0.0001 can be written as \(10^{-4}\) because when we move the decimal point 4 places to the right, we get 1.

Step 3: Apply the Logarithm Property

Using the property of logarithms \(\log_{10}(10^x) = x\), we substitute \(x\) with \(-4\). So, \(\log_{10}(0.0001) = \log_{10}(10^{-4})\).

Step 4: Simplify the Expression

Using the property from the previous step, we find \(\log_{10}(10^{-4}) = -4\). This means the logarithm of 0.0001 with base 10 is -4.

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics. They are a way of expressing repeated multiplication of the same number. For example, when we say \(10^3\), we mean \(10 \times 10 \times 10\). In this case, 10 is the base, and 3 is the exponent, which tells us how many times to multiply 10 by itself.

Using exponents makes it easier to handle very large or very small numbers. Instead of writing 1,000 as a series of three zeros, we can simply say \(10^3\). This notation helps simplify calculations and express numbers more precisely.

Similarly, small numbers like 0.0001 can be expressed using negative exponents. Moving the decimal point four places to the right in 0.0001 gives us the number 1, showing us that \(0.0001 = 10^{-4}\). Exponents, including negative ones, enable us to work effectively with numbers across a wide range of sizes.

Base 10 Logarithm

Base 10 logarithms, often written as \(\log_{10}\), are used to determine the power needed to raise 10 to any given number. In simpler terms, the base 10 logarithm of a number answers the question: "To what power must 10 be raised, to get this number?"

For example, when evaluating \(\log_{10}(1000)\), the answer is \(3\) because \(10^3 = 1000\). Base 10 logarithms are also referred to as common logarithms. They are widely used, especially in scientific calculations where they help with comparing very large or very small measurements.

A key property of logarithms that is useful in simplifying expressions is \(\log_{10}(10^x) = x\). This property allows us to easily find the logarithm of a power of 10, such as when given \(0.0001\) expressed as \(10^{-4}\), leading directly to the conclusion that \(\log_{10}(0.0001) = -4\).

Evaluating Expressions

Evaluating expressions, particularly those involving logarithms and exponents, is a process that requires careful application of mathematical rules and properties. To evaluate \(\log_{10}(0.0001)\), we start by rewriting the number 0.0001 as a power of 10.

Knowing that 0.0001 is equivalent to \(10^{-4}\) helps us apply the logarithmic property \(\log_{10}(10^x) = x\) effectively. This simplifies the expression, allowing us to determine that \(\log_{10}(10^{-4}) = -4\).

When evaluating such expressions, remember these key steps:

• Express the number as a power of 10.
• Use the logarithmic property to simplify the expression.
• Deduce the power to which the base (10) must be raised.

This approach ensures a consistent method of working with logarithms, making complex calculations more approachable and understandable.