Problem 93 - Elementary and Intermediate Algebra

Question

a. \(\log _{2}(x+5)-\log _{2} 4 x=\log _{2} x\) b. \(\ln (x+5)-\ln 4 x=\ln x\)

Step-by-Step Solution

Answer

The solution is \( x = 1.25 \). The \( x = -1 \) is not valid.

Step 1: Use the Quotient Property of Logarithms

The logarithmic identity \( \log_a b - \log_a c = \log_a \left( \frac{b}{c} \right) \) allows us to combine the terms on the left in both equations. For part (a), apply the property: \[ \log_2(x+5) - \log_2(4x) = \log_2 \left( \frac{x+5}{4x} \right) \] For part (b), apply the natural log version: \[ \ln(x+5) - \ln(4x) = \ln \left( \frac{x+5}{4x} \right) \]

Step 2: Equate to the Right-Hand Side

Since both left-hand sides equal \( \log_2 x \) or \( \ln x \) respectively, set equations equal. For (a), equate: \[ \log_2 \left( \frac{x+5}{4x} \right) = \log_2 x \] For (b), equate: \[ \ln \left( \frac{x+5}{4x} \right) = \ln x \]

Step 3: Remove the Logarithms

If the logarithms are equal, the arguments must be equal. Thus for (a) and (b): - For (a): \[ \frac{x+5}{4x} = x \] - For (b): \[ \frac{x+5}{4x} = x \]

Step 4: Solve the Equation for x

Solve the simplified equations for \( x \). Multiply both sides by \( 4x \): - \( x+5 = 4x^2 \) for both (a) and (b), which is the same. Rearrange to form a quadratic equation: - \( 4x^2 - x - 5 = 0 \)

Step 5: Find Roots of the Quadratic

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 4 \), \( b = -1 \), \( c = -5 \). Calculate: \( x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 4 \cdot (-5)}}{2 \cdot 4} \)\( x = \frac{1 \pm \sqrt{1 + 80}}{8} \)\( x = \frac{1 \pm \sqrt{81}}{8} \)\( x = \frac{1 \pm 9}{8} \)The solutions are \( x = 1.25 \) and \( x = -1 \).

Step 6: Check Validity of Solutions

Substitute back into the original equations to check validity. - \( x = 1.25 \): Both log expressions are defined and correct. - \( x = -1 \): Does not satisfy domain of logarithms (not positive).Thus, only \( x = 1.25 \) is valid.

Key Concepts

Quotient Property of Logarithms

The Quotient Property of Logarithms is a powerful tool that simplifies logarithmic expressions. This property states that the difference between two logarithms with the same base can be expressed as a single logarithm. Specifically, if you have

• \( ext{log}_a b - \text{log}_a c = ext{log}_a\left(\frac{b}{c}\right)\)

This allows you to combine multiple logarithmic terms into one, making equations easier to solve. For instance, in the equation \(\log_2(x+5) - \log_2(4x) = \log_2 x\), the left-hand side can be rewritten using the Quotient Property as \(\log_2 \left( \frac{x+5}{4x} \right)\). This results in a simpler expression that can be more readily equated to the single logarithmic term on the right-hand side. Every time you encounter a subtraction of logs with the same base, think about using this property to simplify.

Quadratic Equation

After applying logarithmic properties, you often get simpler expressions, but solving may still require additional steps. In our exercise, simplifying using logarithms eventually leads us to solve a quadratic equation. Quadratics are polynomial equations of degree two, typically in the form \(ax^2 + bx + c = 0\).
When solving such equations, we use the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

In the example, after manipulation, the equation \(4x^2 - x - 5 = 0\) appears. We identified it as a quadratic because it follows the \(ax^2 + bx + c = 0\) pattern. Applying the quadratic formula, we found that this equation gives two potential solutions for \(x\). However, further steps include checking the validity of solutions since not all roots of a quadratic are suitable, especially when they're part of logarithmic functions.

Natural Logarithms

Natural logarithms use the constant \(e\), approximately 2.718, as the base. Instead of writing \(\log_e\), natural logs are denoted as \(\ln\). They are extensively used in natural and applied sciences because they have properties that simplify complex calculations.
The exercise asks us to solve \(\ln (x+5) - \ln (4x) = \ln x\), clearly demonstrating the shift from general logarithms to natural logarithms. This follows similar principles we've seen with other types of logs like using the Quotient Property to merge or split logarithmic terms. In this context:

• \(\ln a - \ln b = \ln\left(\frac{a}{b}\right)\)

helped simplify the equation. Substitution and algebraic manipulation of these forms lead to solving for \(x\), just like non-natural logarithms, but benefit from properties specific to the exponential growth model in natural logs.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. This link means every logarithmic function can be expressed as an exponential equation, which provides a variety of methods for solving. These functions take the form \(y = \log_b(x)\), mapping values from its input to an output that tells you the power to which the base \(b\) must be raised to yield \(x\).
In educational exercises, these skills involve changing complex expressions and equations into simpler ones where both sides often have the same logarithm base. Processing the problem above, after applying logarithmic properties, we equate arguments since the logarithmic property states that if \(\log_b(a) = \log_b(c)\), then \(a = c\). This transforms our logarithmic equation into a solvable algebraic expression.
In real-world applications, logarithmic functions model phenomena like pH in chemistry or the Richter scale in seismology, underscoring the importance of mastering these fundamental mathematical concepts.