Problem 87 - College Algebra and Calculus: An Applied Approach

Question

In Exercises \(87-102\), condense the expression to the logarithm of a single quantity.\(\log _{3} x+\log _{3} 5\)

Step-by-Step Solution

Answer

\(\log _{3} (5x)\)

Step 1: Identify the Property

Recall that the sum of two logarithms with the same base is the logarithm of the product of the numbers. The formula is \(\log _{b} a+\log _{b} c = \log _{b} (ac)\) for any positive numbers a, c, and base b ≠ 1

Step 2: Apply the Property

Applying this property to the expression \(\log _{3} x+\log _{3} 5\), we get the log of the product of the quantities x and 5, with a base of 3.

Step 3: Condense the Expression

The expression condenses to \(\log _{3} (x \cdot 5)\) or, simplifying further, \(\log _{3} (5x)\)

Key Concepts

Understanding Logarithm Rules

Logarithms are an essential concept in mathematics, especially when dealing with exponential functions. The rules of logarithms are mathematical tools that allow us to manipulate and simplify log expressions to make them easier to understand and solve. An important rule is the Product Rule, which states that the sum of two logarithms with the same base is the logarithm of the product of their arguments.

For example, according to the Product Rule, if you have \(\log_b(a) + \log_b(c)\), this can be combined into \(\log_b(ac)\). In the exercise, we applied this rule to condense \(\log_{3} x + \log_{3} 5\) into a single logarithmic expression. Understanding and applying these rules correctly is crucial for working with logarithmic functions in more complex mathematical problems.

Condensing Logarithms Simplified

Condensing logarithms is a process of combining multiple log terms into a single term. It's the inverse process of expanding logarithms. To condense logs effectively, it's essential to be familiar with logarithm properties such as the Product Rule, Quotient Rule, and Power Rule.

For instance, the Product Rule allows us to condense \(\log_{3} x + \log_{3} 5\) into \(\log_{3} (5x)\) by recognizing that the addition of logs with the same base indicates multiplication of their arguments. Condensing logarithms can often lead to a more manageable form for calculations or further algebraic manipulation. Embracing these techniques is an excellent way to enhance problem solving skills in logarithmic functions.

Exploring Logarithmic Functions

A logarithmic function is written as \(y = \log_b(x)\) and is the inverse function of an exponential function of the form \(y = b^x\). Here, 'b' is called the base and must be a positive real number different from 1. Logarithmic functions have unique properties and appear in many scientific applications, including earthquake measurement (Richter scale) and measuring sound intensity (decibels). Understanding logarithmic functions involves becoming comfortable with their graphs, which typically show a curve that passes through the point \(1, 0\) and approaches infinity as 'x' increases.

Additionally, these functions help in solving exponential equations where the unknown variable is an exponent. By transferring the problem into log form, we can utilize logarithm rules to solve for the variable. Knowledge of logarithmic functions is not only vital for algebra but also forms the basis for more advanced topics in calculus and beyond.