Problem 41

Question

For the following exercises, use the one-to-one property of logarithms to solve. $\ln (x-2)-\ln (x)=\ln (54)$

Step-by-Step Solution

Verified

Answer

There is no valid solution since x must be greater than 2.

1Step 1: Use the Logarithm Quotient Rule

Start by applying the quotient rule of logarithms, which states $ \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) $. This simplifies the left-hand side: $ \ln(x-2) - \ln(x) = \ln\left(\frac{x-2}{x}\right) $.

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

According to the one-to-one property of logarithms, if $ \ln(A) = \ln(B) $, then $ A = B $. Apply this property to the equation $ \ln\left(\frac{x-2}{x}\right) = \ln(54) $ to get $ \frac{x-2}{x} = 54 $.

3Step 3: Solve the Equation for x

Now that we have $ \frac{x-2}{x} = 54 $, multiply both sides of the equation by $ x $ to eliminate the fraction, resulting in $ x-2 = 54x $. Rearrange terms to isolate $ x $ on one side: $ x - 54x = 2 $. Simplify to get $ -53x = 2 $.

4Step 4: Isolate x

To isolate $ x $, divide both sides of the equation by $-53$: $ x = \frac{2}{-53} $. This simplifies to $ x = -\frac{2}{53} $. Since $ x = -\frac{2}{53} $ doesn't satisfy the logarithm's domain for $ \, \ln(x) \, $ ($ x > 0 $), we must reconsider and resolve domain issues.

Key Concepts

Logarithm Quotient RuleProperties of LogarithmsSolving Logarithmic Equations

Logarithm Quotient Rule

The Logarithm Quotient Rule is a helpful property that allows you to simplify the difference between two logarithms. In mathematical terms, it states that the difference of two logarithms with the same base, such as $ \ln(a) - \ln(b) $, can be expressed as a single logarithm: $ \ln \left(\frac{a}{b}\right) $. This rule works for any base of logarithms, not just natural logs.Here's why this is useful:

It helps reduce the complexity of an expression by combining multiple logs into one.
This makes it easier to apply further algebraic techniques for problem-solving.
Simplifying logarithmic expressions makes managing equations more efficient.

In the problem $ \ln(x-2) - \ln(x) $, applying the quotient rule simplifies the expression to $ \ln \left( \frac{x-2}{x} \right) $. This simplification is crucial because it sets up the use of the one-to-one property of logarithms.

Properties of Logarithms

Logarithms come with a set of properties that are essential for solving equations, defining their behavior, and simplifying expressions. These properties are particularly useful because they allow for the transformation and manipulation of logarithmic terms.Let's discuss three key properties:

**Logarithm Product Rule**: $ \ln(a) + \ln(b) = \ln(ab) $.
**Logarithm Quotient Rule**: $ \ln(a) - \ln(b) = \ln \left(\frac{a}{b} \right) $.
**Logarithm Power Rule**: $ \ln(a^b) = b \cdot \ln(a) $.

These properties allow you to break down complex logarithmic expressions so they can be easily analyzed and solved with basic algebra. When faced with logarithmic equations, recognizing which property to use can simplify the equation significantly. In the given exercise, the application of the Logarithm Quotient Rule was step one in simplifying the problem to use the one-to-one property efficiently.

Solving Logarithmic Equations

Solving logarithmic equations often involves a series of strategic steps that leverage the properties of logarithms to find the unknown variable. The one-to-one property is especially valuable for this purpose.The one-to-one property asserts that if two logarithmic expressions with the same base are equal, their arguments must also be equal: if $ \ln(A) = \ln(B) $, then $ A = B $. Applying this property simplifies the task of solving logarithmic equations because it converts a logarithmic equation into a simple algebraic one.Here's how you tackle it:

First, simplify all logarithmic expressions using appropriate logarithmic rules.
Then, apply the one-to-one property to eliminate the logarithms.
Finally, solve the resulting algebraic equation to find the solution.

Back to our exercise, the transformation of $ \ln \left( \frac{x-2}{x} \right) = \ln(54) $ into the algebraic equation $ \frac{x-2}{x} = 54 $ showcased this process. Yet, remember: always check domain restrictions since logarithms require positive arguments. In this case, the solution $ x = -\frac{2}{53} $ didn't meet the requirement $ x > 0 $, indicating the need for careful checking of domains to arrive at valid solutions.

Problem 41

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