Problem 504 - Precalculus

Question

Evaluate \(\log (0.000001)\) without using a calculator.

Step-by-Step Solution

Answer

\(-6\)

Step 1: Convert to Exponential Form

First, express 0.000001 in exponential form. Since 0.000001 is equivalent to one millionth, it can be expressed as a power of 10. So, we have: \[ 0.000001 = 10^{-6} \]

Step 2: Apply Logarithm Rules

Use the logarithm power rule: \( \log_b(a^c) = c \cdot \log_b(a) \). Applying this to our problem, we get: \[ \log(0.000001) = \log(10^{-6}) = -6 \cdot \log(10) \]

Step 3: Evaluate Logarithm of 10

We know that the logarithm (common logarithm) of 10 is 1. So, \( \log(10) = 1 \). Substitute this into the expression: \[ -6 \cdot \log(10) = -6 \cdot 1 = -6 \]

Step 4: Final Evaluation

Combine the results from previous steps to obtain the final answer: \[ \log(0.000001) = -6 \]

Key Concepts

Exponential Form

The exponential form is a way of expressing numbers as a base raised to a certain power. This is particularly useful for handling large or small numbers efficiently. In our exercise, the number 0.000001 can be rewritten in exponential form as a power of 10. This involves determining how many times we move the decimal point to transform the number into the base 10 notation. For 0.000001, we shift the decimal 6 places to the right, which translates to:

• 0.000001 becomes 10^-6.

Using exponential form allows for simpler calculations when dealing with logarithms, as you can directly apply logarithm rules to the power of the base.

Logarithm Rules

Logarithm rules are mathematical tools that simplify the process of working with logarithms. One important rule is the power rule, which states:

• \( \log_b(a^c) = c \cdot \log_b(a) \)

This rule lets us move the exponent of the argument to the front as a multiplier. In our exercise, this rule is used to simplify \( \log(10^{-6}) \) to \( -6 \cdot \log(10) \). This transformation streamlines our calculations, turning a potentially complex logarithmic expression into a straightforward multiplication. By mastering these rules, you can handle any logarithmic problem with ease.

Power of 10

The concept of powers of 10 is critical in expressing and understanding different numerical scenarios, especially within scientific and mathematical contexts. Every power of 10 represents a shift in the decimal place.

• Positive powers like 10¹, 10² indicate shifts to the left.
• Negative powers like 10^-1, 10^-2indicate shifts to the right.

For the number 0.000001 in our exercise: converting it to 10^-6 informs us that it is a millionth, as the decimal is shifted 6 places. This is particularly useful when evaluating logarithms, as they often simplify calculations of such exponential representations.

Common Logarithm

A common logarithm is a logarithm with base 10. It is typically written as \( \log(a) \), where the base 10 is implied. This is the type of logarithm most commonly used in scientific contexts and is particularly standard for calculations involving powers of 10.

• One crucial fact is that \( \log(10) = 1 \).

In our example, understanding that the common logarithm of 10 is 1 allows us to simplify our problem. So, when computing \( \log(10^{-6}) \), using the establishment that \( \log(10) = 1 \), results in using the power rule efficiently, confirming the simplification yields \( -6 \cdot 1 = -6 \). Mastery of common logarithms is essential for efficient calculation and problem-solving in logarithmic contexts.