Problem 9 - College Algebra Essentials

Question

Use properties of logarithms to expand 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 form of the logarithmic expression \(\log \left(\frac{x}{100}\right)\) is \(\log(x) - 2\).

Step 1: Identify the applicable logarithmic property

The problem can be solved by applying the quotient rule of logarithms, which states that for any positive numbers a, b (b ≠ 1) and x > 0, we have \(\log_b \left(\frac{a}{x}\right) = \log_b(a) - \log_b(x)\).

Step 2: Apply the logarithmic property

Using this property on the given expression \(\log \left(\frac{x}{100}\right)\) simplifies it to \(\log(x) - \log(100)\).

Step 3: Evaluate the logarithmic expressions

The logarithm base 10 of 100 is 2, because 10 raised to the power of 2 gives 100. Substitute it into the expression, we can then simplify the expression as \(\log(x) - 2\).

Key Concepts

Quotient Rule

The quotient rule is a fundamental property of logarithms that simplifies the division of two numbers within a logarithmic expression. It states that for any positive numbers \(a, b,\) and \(x\) (where \(b eq 1\) and \(x > 0\)), the logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator. In formulaic terms: \[\log_b \left(\frac{a}{x}\right) = \log_b(a) - \log_b(x)\]This rule essentially allows us to break down complex fractions into simpler, manageable parts. Applying this to the expression \(\log \left(\frac{x}{100}\right)\), we break it into \(\log(x) - \log(100)\). This makes it easier for you to calculate or further expand the expression without complex computation.
Using the quotient rule helps reduce errors and makes solving logarithmic problems more straightforward. Make sure you fully understand this property as it is widely used in solving logarithmic expressions.

Logarithmic Expressions

Logarithmic expressions involve the use of logarithm properties to simplify, expand, or solve equations. Understanding how to manipulate these expressions using logarithmic properties is crucial. The properties include the product rule, quotient rule, power rule, and change of base formula. Each has its specific use case to simplify different mathematical scenarios.For instance, to expand \(\log \left(\frac{x}{100}\right)\), recognizing that it is a quotient within a logarithm expression allows us to use the quotient rule. By doing this, it becomes clear that the expression can be expanded to \(\log(x) - \log(100)\). Once expanded, these components can be evaluated independently if possible.
When working through logarithmic expressions, breaking them down systematically using these rules enables cleaner solutions and often illuminates paths to simplification. Practice regularly with different types of expressions to become comfortable handling any logarithmic problem.

Base 10 Logarithms

Base 10 logarithms, often referred to as common logarithms, are logarithms that use 10 as the base. It is denoted as \(\log\) rather than \(\log_{10}\). Understanding base 10 is crucial, as it is widely used in scientific and engineering calculations.Consider the problem expression \(\log(100)\). Since 100 is equal to \(10^2\), using base 10 logarithm, \(\log(100)\) simplifies to 2 because 10 raised to the power of 2 equals 100. Recognizing such simplifications is key when working with logarithmic expressions.

• Many calculators are set to compute base 10 logs directly.
• Base 10 is intuitive for numbers expressed in scientific notation.

This makes solving the expression \(\log(x) - 2\) straightforward, where \(\log(100)\) has been calculated independently and subtracted from \(\log(x)\). Base 10 logarithms simplify many otherwise complex calculations, making them an essential part of mathematics.