Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { In Problems 17-24, solve for } x . \text { Hint: } \log _{a} b=c \Leftrightarrow a^{c}=b \text {. } $$ $$ \log _{5}(x+3)-\log _{5} x=1 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{3}{4} \).

Step 1: Apply Logarithm Properties

We begin by using the property of logarithms: \( \log_a b - \log_a c = \log_a \left( \frac{b}{c} \right) \). Thus, the expression given is \( \log_5 \left( \frac{x+3}{x} \right) = 1 \).

Step 2: Convert the Logarithm Equation to Exponential Form

Using the hint that \( \log_a b = c \) is equivalent to \( a^c = b \), we rewrite the equation \( \log_5 \left( \frac{x+3}{x} \right) = 1 \) as \( 5^1 = \frac{x+3}{x} \).

Step 3: Solve the Exponential Equation

The equation \( 5 = \frac{x+3}{x} \) implies that \( 5x = x + 3 \). To find \( x \), we solve for \( x \):\[ 5x = x + 3 \] \[ 5x - x = 3 \] \[ 4x = 3 \] \[ x = \frac{3}{4} \]

Step 4: Verify the Solution

Substitute \( x = \frac{3}{4} \) back into the original equation to verify:\( \log_5\left( \frac{\frac{3}{4} + 3}{\frac{3}{4}} \right) = \log_5(5) = 1 \).This confirms the solution is correct as it satisfies the equation.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is crucial when dealing with equations that involve logs. One essential property used in solving logarithmic equations is the difference of logarithms rule: \( \log_a b - \log_a c = \log_a \left( \frac{b}{c} \right) \). This property allows us to combine two logarithmic expressions into one
simplification. By using this property, we can transform the equation into a more manageable form, paving the way for simplification and solving of the variables involved.

Here's a quick rundown of some other key properties that often come in handy:

• Logarithm of a product: \( \log_a (bc) = \log_a b + \log_a c \)
• Logarithm of a quotient: \( \log_a \left( \frac{b}{c} \right) = \log_a b - \log_a c \)
• Logarithm of a power: \( \log_a (b^c) = c \log_a b \)

These properties are instrumental when simplifying logarithmic expressions and solving equations involving logarithms.

Exponential Form

When solving logarithmic equations, converting them to exponential form can be an efficient strategy. The fundamental idea is rooted in the relationship between logarithms and exponents. If you have an equation of the form \( \log_a b = c \), it can be rewritten or "decoded" into exponential form as \( a^c = b \).

This conversion is immensely helpful because it simplifies the logarithmic equation into a typical algebraic equation that is often much easier to solve. In the exercise presented, the equation \( \log_5 \left( \frac{x+3}{x} \right) = 1 \) is transformed into its exponential form \( 5^1 = \frac{x+3}{x} \), which simplifies the solving process significantly.
Understanding this conversion can deepen your comprehension of the inherent relationship between exponential and logarithmic operations, and it is a fundamental technique in tackling complex equations.

Solving Equations

Solving equations, particularly those that arise from logarithmic expressions, can sometimes seem daunting. However, with a structured approach, it becomes much more manageable. Once you've simplified the logarithmic expression using properties of logarithms and converted it into exponential form, the task is reduced to solving a basic algebraic equation.

To solve equations like \( 5 = \frac{x+3}{x} \), the goal is to isolate \( x \). Here are the steps that follow:

• Multiply both sides by the denominator to get rid of the fraction, resulting in \( 5x = x + 3 \).
• Simplify and isolate the variable, \( x \), by moving all terms involving \( x \) to one side.
• Combine and solve for \( x \), which leads us to \( x = \frac{3}{4} \).

Following these logical steps helps ensure a neat and successful resolution to the problem.

Verification of Solutions

Once an equation is solved, verifying the solution ensures that no mistakes were made and that the answer satisfies the original problem. This step is essential in confirming the reliability and correctness of your solution.

To verify, simply substitute \( x = \frac{3}{4} \) back into the original logarithmic equation to see if it holds true:

• Substitute \( x \) in: \( \log_5 \left( \frac{\frac{3}{4} + 3}{\frac{3}{4}} \right) \)
• Simplify to get \( \log_5(5) \) which equals 1, confirming the left side matches the given right side.

This substitution ensures that the initial assumptions and solved answers are indeed compliant with the original equation, safeguarding against errors in calculation. Always double-check your solution in the original context to verify its accuracy.