Problem 59 - Elementary and Intermediate Algebra

Question

Solve each equation. See Example \(9 .\) $$ \log _{3} 4 x-\log _{3} 7=2 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{63}{4} \) or 15.75.

Step 1: Use Logarithmic Properties

The given equation is \( \log_{3} (4x) - \log_{3} (7) = 2 \). According to the properties of logarithms, specifically the quotient property, we can combine the logs: \( \log_{3}\left(\frac{4x}{7}\right) = 2 \).

Step 2: Convert Logarithmic Equation to Exponential Form

Now that we have \( \log_{3}\left(\frac{4x}{7}\right) = 2 \), we convert it to exponential form. Recall that if \( \log_{b}(a) = c \), then \( b^c = a \). So, \( 3^2 = \frac{4x}{7} \). This simplifies to \( 9 = \frac{4x}{7} \).

Step 3: Solve for x

To solve for \( x \), multiply both sides of the equation \( 9 = \frac{4x}{7} \) by 7: \( 9 \times 7 = 4x \), which simplifies to \( 63 = 4x \). Divide both sides by 4 to isolate \( x \): \( x = \frac{63}{4} \).

Step 4: Simplify the Result

The simplified form of \( x = \frac{63}{4} \) can be left as an improper fraction: \( x = \frac{63}{4} \) or written as a mixed number, \( x = 15.75 \).

Key Concepts

Solving Equations

Solving equations is a fundamental aspect of mathematics and is crucial in many areas like science, engineering, and everyday problem-solving. In this exercise, the task is to find the value of \( x \) that satisfies the equation \( \log_{3}(4x) - \log_{3}(7) = 2 \). The first step often involves simplifying or manipulating the equation to a more manageable form using various mathematical properties. This allows us to systematically determine what \( x \) equals.

• Start by simplifying the equation using known mathematical properties, such as the properties of logarithms in this case.
• Continue by expressing the equation in a way that makes extracting the variable straightforward.

Once simplified, the equation or inequality may be solved by performing operations that isolate the variable \( x \). This often involves a series of algebraic manipulations like adding, subtracting, multiplying, dividing, or even using substitution.

Logarithmic Properties

Logarithms are an exciting mathematical concept that makes complex calculations simpler by transforming multiplication into addition and powers into products. When solving equations with logarithms, it's essential to leverage logarithmic properties. In this problem, the original equation given is \( \log_{3}(4x) - \log_{3}(7) = 2 \).

Using the Quotient Property

One of the crucial properties of logarithms is the quotient property, which states \( \log_{b}\left(a\right) - \log_{b}(c) = \log_{b}\left(\frac{a}{c}\right)\). Using this property, we can simplify the equation to \( \log_{3}\left(\frac{4x}{7}\right) = 2 \).

Benefits of Simplification

By applying the quotient property, we get a single log expression, making it easier to solve. This simplification is not just a technical step but a strategic one, transforming potential complexity into a path toward a solution.

Exponential Form

Once the logarithmic equation is simplified, converting it to exponential form can be a very effective step. In our exercise, after applying the logarithmic properties, we have \( \log_{3}\left(\frac{4x}{7}\right) = 2 \).

Understanding the Conversion

The definition of logarithms allows us to transform this equation into its equivalent exponential form. Remember, if \( \log_{b}(a) = c \), then \( b^c = a \). This leaves us with \( 3^2 = \frac{4x}{7} \), which simplifies to \( 9 = \frac{4x}{7} \).

Why Use Exponential Form?

Converting to exponential form simplifies the process because the equation now only involves basic arithmetic. It allows one to solve for the variable \( x \) using simple operations such as multiplication and division. This step bridges the gap from a logarithmic equation to a solvable equation, clarifying the path to the solution.