Problem 74 - Precalculus with Limits

Question

Use the One-to-One Property to solve the equation for \(x .\) \(\ln (x-7)=\ln 7\)

Step-by-Step Solution

Answer

The solution to the equation \(\ln (x-7)=\ln 7\) is \(x = 14\).

Step 1: Identify the Problem

You are given the equation \(\ln (x-7)=\ln 7\). Notice that both sides of the equation have a logarithm with the same base (natural log or \(\ln\)), making it possible to use the one-to-one property of logarithms.

Step 2: Apply the One-to-One Property of Logarithms

By the one-to-one property of logarithms, if \(\ln a = \ln b\), then \(a = b\). So in our equation \(\ln (x-7)=\ln 7\), we can say that \(x - 7 = 7\).

Step 3: Isolate the variable

Next, add 7 to both sides of the equation to solve for \(x\). This gives \(x = 7 + 7 = 14\).

Key Concepts

Logarithms

In mathematics, logarithms are a way to express exponents or powers.They help in turning multiplication into addition, which is often easier to work with.When you see a logarithmic equation, it usually involves finding a power that a number must be raised to, to get another number.

For instance, if you have a base of 10 (common logarithm), and you want to know what power to raise it to get 100, you would use \( ext{log}_{10} 100 = 2\).This tells us that 10 raised to the power of 2 is 100.Logarithms are important in solving equations that deal with exponential growth or decay.

In the context of the original exercise, we work with a natural logarithm, denoted \(\ln\).But the concept of how logarithms simplify complicated problems remains the same.

Natural Logarithm

A natural logarithm, often denoted by \(\ln\), is a special type of logarithm where the base is the mathematical constant \(e\).This constant is approximately 2.71828 and arises naturally in many areas of mathematics, especially those involving growth processes.

Using natural logarithms helps to solve problems involving continuous growth or decay.The \(\ln\) function is the inverse of the exponential function with base \(e\), meaning that \(\ln(e^x) = x\) for any real number \(x\).In our original problem, both sides of the equation involved the natural logarithm, making it feasible to use the one-to-one property.

Natural log equations are especially common in calculus, physics, and engineering domains, offering a robust tool for mathematical analysis.

Equation Solving

Solving equations often involves finding the value of the unknown variable that makes the equation true.In the original problem, this involved solving for \(x\) using the one-to-one property of logarithms.

This property states that if \(\ln a = \ln b\), then we can safely conclude \(a = b\).This is because logarithmic functions with the same base are injective, meaning they map different inputs to different outputs.Thus, knowing the two outputs are equal assures us that the inputs also must be the same.

After applying the one-to-one property, you need to proceed to isolate the variable.This often involves performing basic arithmetic operations to both sides of the equation until the variable stands alone.