Problem 66 - A Graphical Approach to Precalculus with Limits

Question

Explain why this logarithmic equation has no solution. $$\ln x-\ln (x+1)=\ln 5$$

Step-by-Step Solution

Answer

The equation has no solution because the logarithms require positive arguments, but \( x = -\frac{5}{4} \), which is not positive.

Step 1: Simplify the Logarithmic Expression

To start, we use the logarithmic property where \( \ln a - \ln b = \ln \left(\frac{a}{b}\right) \). Applying this to the expression \( \ln x - \ln (x+1) \), we get:\[ \ln \left( \frac{x}{x+1} \right) = \ln 5 \].

Step 2: Exponentiate Both Sides

Next, exponentiate both sides to eliminate the natural logarithm. This gives us:\[ \frac{x}{x+1} = 5 \].

Step 3: Solve the Resulting Equation

Now, solve for \( x \) by multiplying both sides by \( x+1 \) to clear the fraction:\[ x = 5(x + 1) \]Simplify to get:\[ x = 5x + 5 \].

Step 4: Rearrange and Simplify the Equation

Subtract \( 5x \) from both sides to get:\[ x - 5x = 5 \]which simplifies to:\[ -4x = 5 \].

Step 5: Solve for x

To isolate \( x \), divide both sides by \(-4\):\[ x = -\frac{5}{4} \].

Step 6: Check the Validity of the Solution in the Original Equation

Logarithms are only defined for positive numbers, thus we need \( x > 0 \) and \( x+1 > 0 \). Since \( x = -\frac{5}{4} \) is not greater than 0, the solution does not satisfy the domain of the logarithm function. This means the original logarithmic equation has no solutions.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a specific logarithm with the base \( e \), where \( e \approx 2.71828 \). It is a fundamental function in mathematics, especially in calculus, due to its simple derivation properties and appearances in formulas related to continuous growth or decay. Natural logarithms are only defined for positive real numbers. This means that for any expression involving \( \ln x \), \( x \) must be greater than zero. In the context of the logarithmic equation \( \ln x - \ln (x+1) = \ln 5 \), the values of \( x \) must satisfy the condition of being positive and also ensuring \( x + 1 > 0 \), which further assures \( x > -1 \). However, since \( x \) is derived as a negative number from solving the equation, it violates the domain requirement of the natural logarithm.

Properties of Logarithms

Understanding the properties of logarithms is crucial when working with logarithmic equations. These properties allow us to simplify and solve equations effectively. One key property is the difference of logarithms known as the quotient rule: \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \). Applying this property to simplify the equation \( \ln x - \ln (x+1) = \ln 5 \) results in \( \ln \left( \frac{x}{x+1} \right) = \ln 5 \). This property substantially simplifies the equation by reducing the number of logarithmic terms, making it easier to solve. Another important property involves exponentiation, which helps to clear logarithms from both sides of an equation. Remember, applying these rules means checking that the resulting expressions respect the domain of logarithms: all inputs to a natural log must remain positive.

Exponentiation in Equations

Exponentiation is a technique often used to solve equations involving logarithms. By raising both sides of a logarithmic equation to the power of the base of the logarithm, we can cancel out the logarithms. In this problem, exponentiating both sides of the equation \( \ln \left( \frac{x}{x+1} \right) = \ln 5 \) essentially translates it to \( \frac{x}{x+1} = 5 \). This step removes the natural logarithms, transforming the equation into a simpler algebraic form.When dealing with such transformations, it's crucial to maintain the integrity of the variable’s conditions. In our case, solving the new equation gives \( x = -\frac{5}{4} \). Even though the arithmetic is correct, this value does not fit the natural logarithm's domain restrictions mentioned previously, thus confirming that no solution exists for the initial equation.