Problem 31 - Algebra for College Students

Question

Evaluate each logarithmic expression. \(\log _{10} 0.1\)

Step-by-Step Solution

Answer

\( \log_{10} 0.1 = -1 \).

Step 1: Understand the Logarithmic Expression

The expression given is \( \log_{10} 0.1 \), which denotes the logarithm of 0.1 with base 10. This means we need to find the power to which 10 must be raised to get 0.1.

Step 2: Set the Expression Equal to a Variable

Let \( x = \log_{10} 0.1 \). This means that \( 10^x = 0.1 \). We need to solve for \( x \) to find the value of the logarithm.

Step 3: Express the Base Number as a Power

Recognize that 0.1 can be written as \( \frac{1}{10} \), which is the same as \( 10^{-1} \). Therefore, the equation becomes \( 10^x = 10^{-1} \).

Step 4: Solve the Equation

Since the bases are the same, we can set the exponents equal to each other. Thus, \( x = -1 \).

Step 5: Verify the Solution

Check that \( 10^{-1} = 0.1 \), which verifies that \( x = -1 \) is correct. Therefore, \( \log_{10}(0.1) = -1 \).

Key Concepts

Base 10 Logarithms

The concept of logarithms can sometimes feel a bit abstract at first, but it's all about finding what power you need to raise a certain number, the base, to get another number. When we talk about "base 10 logarithms," we're working with a specific type of logarithm where the base is 10. This is commonly used in scientific notation and is also known as the "common logarithm."
The question we often ask with base 10 logarithms is: To what exponent must 10 be raised to result in a given number? For example, if we have the expression \( \log_{10} 100 \), we're essentially asking what power we need to raise 10 to in order to get 100. The answer, of course, is 2, because \( 10^2 = 100 \).
Writing logarithms using base 10 helps simplify calculations and allows us to move between multiplication and exponentiation rather seamlessly. Logarithms compress information, turning large products into manageable sums.

Exponents and Powers

Understanding exponents is crucial when dealing with logarithms because they are deeply interlinked. An exponent denotes how many times a number, known as the base, is multiplied by itself. For instance, the expression \( 10^3 \) indicates that 10 is multiplied by itself three times, which equals 1,000.
Working with powers requires grasping a few essential rules:

• Any number to the power of zero is 1, e.g., \( 10^0 = 1 \).
• A negative exponent implies division or a fraction, e.g., \( 10^{-1} = \frac{1}{10} \).
• A fractional exponent represents a root, e.g., \( 10^{0.5} = \sqrt{10} \).

In our exercise, understanding that \( 0.1 \) can be rewritten as \( 10^{-1} \) helps solve the problem efficiently. Recognizing such patterns is a key mathematical skill that simplifies more complex logarithmic and exponential calculations.

Solving Logarithmic Equations

Solving a logarithmic equation often involves finding what exponent or power a base number should be raised to, in order to achieve the desired result. You typically start by setting the logarithmic equation equal to a variable, which makes it easier to manipulate mathematically.
For instance, when we solve \( \log_{10} 0.1 \), we set it equal to \( x \) (i.e., \( x = \log_{10} 0.1 \)), which implies that \( 10^x = 0.1 \). Our task now is to reveal the value of \( x \). This requires expressing \( 0.1 \) as a power of 10. Knowing that \( 0.1 = 10^{-1} \), we can simply say \( 10^x = 10^{-1} \) and thus \( x = -1 \).
The final step should always be verification. By substituting \( x \) back into the equation, you confirm the result. In this case, checking that \( 10^{-1} \) indeed equals 0.1 verifies the solution is correct. Such verification ensures precision in mathematical problem-solving and builds a strong foundational understanding.