Problem 160 - Precalculus

Question

For the following exercises, use the definition of common and natural logarithms to simplify. $$2 \log (.0001)$$

Step-by-Step Solution

Answer

The simplified result is -8.

Step 1: Understand the logarithm

The common logarithm is a logarithm to the base 10, denoted as \(\log\). It measures the power to which the number 10 must be raised to obtain a certain value.

Step 2: Rewrite the decimal as a power of 10

Recognize that 0.0001 can be written as \(10^{-4}\). This simplifies logarithmic calculations.

Step 3: Apply the logarithmic identity

Use the property \( \log(a^b) = b\cdot\log(a) \). Thus, \( \log(0.0001) = \log(10^{-4}) = -4 \cdot \log(10) \).

Step 4: Simplify with the known value

Recall that \( \log(10) = 1 \). Therefore, \(-4 \cdot \log(10) = -4 \cdot 1 = -4\).

Step 5: Multiply by the given constant

Now, compute \( 2 \cdot \log(.0001) = 2 \cdot (-4) = -8\).

Key Concepts

Common Logarithms

A common logarithm is simply a logarithm with base 10. It's denoted as \(\log\) and often used in calculations when dealing with numbers less than or greater than 10. This is especially true in scientific fields, where measuring scales like pH levels or sound intensity are expressed logarithmically.

• Key fact: \(\log(10) = 1\), because 10 raised to the power of 1 equals 10.
• Think of it like asking "To what power must I raise 10 to get my desired number?"

Common logarithms simplify expressing and solving exponential relationships involving base 10. Whenever you see \(\log\) without a base, assume the base is 10. This helps in simplifying expressions and solving equations efficiently.

Properties of Logarithms

Logarithms have unique properties that make calculations easier and more manageable.

• Power Rule: \(\log(a^b) = b \cdot \log(a)\). This allows you to take the exponent in front as a multiplier.
• Product Rule: \(\log(a \cdot b) = \log(a) + \log(b)\).
• Quotient Rule: \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\).

These properties help in transforming complex logarithmic expressions into simpler forms. In the exercise, transforming \(0.0001\) into its power of 10 form \(10^{-4}\) was crucial. By applying the power rule, the negative exponent became a coefficient, which was then easy to handle.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves using logarithmic properties to transform a complex term into more manageable parts. Let's look at the steps applied in the exercise:

• Convert decimals or other forms into powers of 10 to make use of common logarithms.
• Utilize the power rule to bring the exponent out as a factor.
• Combine like terms and perform basic arithmetic operations.

For example, simplifying \(2\log(0.0001)\) involves rewriting 0.0001 as \(10^{-4}\). Employ the power rule so that \(\log(10^{-4}) = -4\cdot\log(10)\), and subsequently finding \(-4\). Finally, multiply by 2 as given, leading to the result of \(-8\). These transformation steps clarify the originally daunting logarithmic expression, revealing simplicity in what first seemed complex.