Problem 49 - College Algebra

Question

For the following exercises, evaluate the common logarithmic expression without using a calculator. $$2 \log \left(100^{-3}\right)$$

Step-by-Step Solution

Answer

The expression evaluates to -12.

Step 1: Understand the problem

We need to evaluate the expression \(2 \log \left(100^{-3}\right)\) without using a calculator. This involves applying logarithmic rules to simplify the expression.

Step 2: Apply power rule of logarithms

The power rule of logarithms states that \( \log(b^a) = a \log(b) \). Apply this rule to \( \log \left(100^{-3}\right) \) as follows: \( \log \left(100^{-3}\right) = -3 \log(100) \).

Step 3: Substitute and simplify

Substitute the result from Step 2 into the original expression: \[2 \log \left(100^{-3}\right) = 2 (-3 \log(100))\]Simplify to get \[-6 \log(100)\]

Step 4: Evaluate the logarithm

Recall that the common logarithm \(\log(100)\) (logarithm base 10) is 2, since 100 is \(10^2\). Substitute this into the expression: \[-6 \times 2 = -12\]

Step 5: Conclude the solution

The expression evaluates to \(-12\). Thus, \(2 \log \left(100^{-3}\right) = -12\).

Key Concepts

Logarithmic Rules

Logarithmic rules are fundamental principles that allow us to simplify and solve logarithmic expressions. These rules include several handy properties such as the product rule, quotient rule, and power rule, which help in manipulating expressions to reach the desired solution.

• The **Product Rule** states that the logarithm of a product is equal to the sum of the logarithms of the factors, expressed as \(\log(ab) = \log(a) + \log(b)\).
• The **Quotient Rule** tells us that the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator: \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\).
• The **Power Rule** indicates that the logarithm of an exponential expression can be brought down as a multiplication, given as \(\log(b^a) = a\log(b)\). This is especially useful for simplifying expressions where the variable or base has an exponent.

These rules are incredibly useful for simplifying expressions without the need for complex calculations or a calculator, as demonstrated in the given problem.

Common Logarithm

The common logarithm, often symbolized as \(\log(x)\), is the logarithm with base 10. This is widely used in many fields including science and engineering because it simplifies many calculations by returning results that are relevant in a decimal system which is natural for human interpretation.

The common logarithm has specific values that are important to remember, especially for foundational numbers:

• \(\log(10) = 1\), because 10 is 10 raised to the power of 1.
• \(\log(100) = 2\), as 100 is 10 raised to the power of 2.
• \(\log(1000) = 3\), since 1000 is 10 raised to the power of 3.

These specific values can quickly help you evaluate expressions mentally, especially when dealing with powers of 10, eliminating the need for calculators. This is what made it possible to simplify our original exercise to the answer \(-12\).

Power Rule of Logarithms

The power rule of logarithms is one of the most powerful rules used to simplify logarithmic expressions. It states that \(\log(b^a) = a\log(b)\), allowing one to "bring down" the exponent to the front of the logarithm as a multiplier. This transformation is crucial for tackling complex expressions.

Consider how in the provided solution, the term \(\log(100^{-3})\) was simplified to \(-3\log(100)\). Here, the exponent \(-3\) was moved out in front of the logarithm, making it straightforward to apply other rules or directly substitute values.

This rule is particularly useful because logarithmic values can be known or estimated for many bases, and the exponent simplifies the evaluation step significantly.

• It's a time-saver for integrating logarithmic expression solving into broader equations.
• Helps keep calculations manageable and accurate, especially during exams or quick calculations.

Understanding and applying the power rule effectively can turn an otherwise cumbersome log problem into a quick and easy computation.