Problem 9 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log \left(\frac{x}{100}\right) $$

Step-by-Step Solution

Answer

The expanded and simplified form of the given logarithmic expression is \( \log(x) - 2\).

Step 1: Identify the law of logarithms to apply

Consider the given logarithm \( \log \left(\frac{x}{100}\right) \). Here, the law of logarithm that applies is: \(\log(a / b) = \log(a) - \log(b)\). This can be used because the logarithm contains a division operation inside it.

Step 2: Apply the law of logarithm

Applying the law of logarithm from step 1, the given logarithmic expression expands to: \( \log(x) - \log(100)\).

Step 3: Simplify the logarithmic expression

The \(\log(100)\) can be evaluated without a calculator because \(10^2 = 100\). Hence, it simplifies to 2. Therefore, the original logarithmic expression simplifies to: \( \log(x) - 2\).

Key Concepts

Logarithmic Expressions

Logarithmic expressions are mathematical statements that involve the logarithm function, indicative of an exponential relationship. The core of understanding a logarithmic expression lies in knowing that a logarithm, denoted as \( \log_b(a) \), answers the question: 'To what power must we raise the base \( b \) to obtain \( a \)?' When dealing with such expressions, several properties of logarithms, such as product, quotient, and power rules, become essential tools.

For example, \( \log(\frac{x}{100}) \) is a logarithmic expression that indicates the power to which 10 must be raised to get the fraction \( \frac{x}{100} \). Remember, the base of \( \log \) without a subscript is commonly assumed to be 10. This expression contains a division, which directly leads us to consider the quotient property for simplification.

Expand Logarithms

To expand logarithms means to express a solitary logarithmic expression as the difference, sum, or product of simpler logarithms. The operation often utilizes three fundamental properties: the Product Rule, Quotient Rule, and Power Rule.

The Quotient Rule dictates that \( \log(\frac{a}{b}) \) is equivalent to \( \log(a) - \log(b) \), which is applicable when you have a division inside the logarithm, as we see in the expression \( \log(\frac{x}{100}) \). By expanding logarithms, you break down complex log expressions into more manageable parts, often making it easier to evaluate or simplify further.

Simplify Logarithms

Simplifying logarithms involves reducing a logarithmic expression into its simplest form using logarithmic identities and properties. After expanding a logarithm, you often get terms that can be further simplified. For instance, a term like \( \log(100) \) can be recognized as \( \log(10^2) \), which simplifies to 2 because the base of the logarithm is 10.

Thus, the process not only makes the expression more straightforward but also prepares it for evaluation, should that be your next step. Key properties used in simplifying logarithms include turning products into sums, quotients into differences, and exponents into multiplications by the log's base.

Evaluate Logarithms without a Calculator

Evaluating logarithms without a calculator is an essential skill, testing one's understanding of how logarithms reflect exponential relationships. This often involves recognizing when the argument of a logarithm is a power of the base, as is the case with \( \log(100) \), where 100 is \( 10^2 \). Knowing that the base of \( \log \) is 10 allows us to evaluate this logarithm as 2.

Sometimes, using the change of base formula or mental math for familiar power relationships helps evaluate more complicated logarithms. The goal is to transform the logarithmic expression into a form involving integers or known values, thus avoiding the need for computational tools.