Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$ \log x-\log (x+1)=3 \log 4 $$

Step-by-Step Solution

Answer

The equation has no solution because the found \( x \) is negative, making logarithms undefined.

Step 1: Use Logarithmic Identity

We start by using the property of logarithms that states \( \log a - \log b = \log \left( \frac{a}{b} \right) \). Applying this to the left side of the equation, we transform \( \log x - \log (x+1) \) into \( \log \left( \frac{x}{x+1} \right) \). Our equation becomes: \[ \log \left( \frac{x}{x+1} \right) = 3 \log 4 \]

Step 2: Apply Power Rule for Logarithms

Using the power rule of logarithms, which states \( a \log b = \log b^a \), we rewrite the right side of the equation. Thus, \( 3 \log 4 = \log 4^3 \). Simplifying, we know \( 4^3 = 64 \), so the equation becomes: \[ \log \left( \frac{x}{x+1} \right) = \log 64 \]

Step 3: Equate Arguments

Since the logs are equal, the arguments must be equal. Thus, we have: \[ \frac{x}{x+1} = 64 \]

Step 4: Solve the Rational Equation

To eliminate the fraction, multiply both sides by \( (x+1) \), giving: \[ x = 64(x + 1) \] Distribute the \( 64 \) on the right-hand side: \[ x = 64x + 64 \]

Step 5: Isolate x

Subtract \( 64x \) from both sides to get all terms involving \( x \) on one side: \[ x - 64x = 64 \] Simplify to: \[ -63x = 64 \] Divide both sides by \( -63 \): \[ x = -\frac{64}{63} \]

Step 6: Check and Interpret the Solution

Since negative values of \( x \) would make the domain of the original logarithmic functions invalid, and logarithms require positive arguments, the equation \( \log x \) is undefined for \( x = -\frac{64}{63} \). Thus, there is no valid solution for \( x \).

Key Concepts

Properties of Logarithms

Logarithms come with a set of useful properties that help us simplify calculations and solve equations efficiently. One of the key properties is the **Difference Identity**, which states that the difference of two logarithms can be expressed as a single logarithm of their quotient. Mathematically, this is given by: \( \log a - \log b = \log \left( \frac{a}{b} \right) \). This property is particularly valuable when solving equations, as it allows us to combine terms and simplify the expression of logarithmic equations.

Another essential property is the **Power Rule**. It allows us to move the coefficient of the logarithm into the argument as an exponent. This is written as \( a \log b = \log b^a \). This rule is incredibly helpful when dealing with exponential expressions within a logarithmic function, as it paves the way for simplification and solving the equations.

These properties are fundamental when working with logarithmic functions and assist in streamlining the problem-solving process, making complex equations more manageable.

Solving Rational Equations

Solving rational equations involves finding the values of the variable that make the equation true. These equations include fractions and the variable, often in the denominators. The typical approach to solving them includes two main steps: **Finding a Common Denominator** and **Eliminating Fractions**. In the original solution, we handled the rational expression \( \frac{x}{x+1} = 64 \), which required eliminating the fraction.

To do so, we multiplied both sides by \( x + 1 \), removing the fraction and leading us to a simpler equation. This step is critical, as it ultimately allows you to deal with standard linear or polynomial equations without fraction complications. However, a key part of solving such equations involves checking the solutions to ensure they fall within the domain of validity, as unforeseen restrictions might arise from the original conditions.

Logarithmic Identities

Logarithmic identities are tools that help manipulate and transform logarithmic expressions for easier computation or to fit a desired form. They are indispensable in resolving equations involving logarithms. Some common identities include:

• **Product Identity**: \( \log(ab) = \log a + \log b \)
• **Quotient Identity**: \( \log \left( \frac{a}{b} \right) = \log a - \log b \)
• **Power Identity**: \( \log a^b = b \log a \)

These identities often underpin the simplification process in solving logarithmic equations. As seen in the example, the application of these identities allows us to progress through and ultimately solve the equation by equating both sides, once they share a common log argument. Knowing these identities enhances one's ability to deconstruct logarithmic problems efficiently.

Domain of Logarithmic Functions

The domain of a logarithmic function is the set of all permissible values for which the function is defined. Since logarithms are undefined for zero or negative arguments, it's crucial to ensure that within any expression involving logarithms, each argument stays positive.

For the equation \( \log x - \log (x+1) = 3 \log 4 \), the domain restriction means both \( x \) and \( x+1 \) must be greater than zero. This imposes a natural boundary on the solutions we should consider. Specifically, \( x > 0 \) and \( x+1 > 0 \) must hold, simplifying to \( x > 0 \).

In the case at hand, the solution derived was \( x = -\frac{64}{63} \), a negative number, making it invalid within the domain of the original equation. Understanding domain constraints ensures that solutions are not only algebraically valid but also meaningful within the context of the function characteristics.