Problem 37 - College Algebra

Question

For the following exercises. use the one-to-one pronerty of logarithms to solve. $$ \log _{13}(5 n-2)=\log _{13}(8-5 n) $$

Step-by-Step Solution

Answer

The solution to the equation is \(n = 1\).

Step 1: Understand the One-to-One Property of Logarithms

The one-to-one property of logarithms states that if \( \log_a(x) = \log_a(y) \), then \( x = y \). This means that if the logarithms of two expressions with the same base are equal, then the expressions themselves must be equal.

Step 2: Set the Arguments Equal

Using the one-to-one property of logarithms, we can set the arguments of the logarithms equal to each other: \[ 5n - 2 = 8 - 5n \]

Step 3: Solve the Linear Equation

We have the equation \(5n - 2 = 8 - 5n\). To solve for \(n\), first add \(5n\) to both sides: \[ 5n + 5n - 2 = 8 \]This simplifies to \[ 10n - 2 = 8 \] Next, add 2 to both sides: \[ 10n = 10 \] Finally, divide both sides by 10 to isolate \(n\): \[ n = 1 \]

Key Concepts

One-to-One Property of Logarithms

The one-to-one property of logarithms is a powerful tool for solving logarithmic equations. This property simplifies complex problems by allowing us to equate the arguments of logarithms that share the same base. In essence, if you have an equation like \( \log_a(x) = \log_a(y) \), you can directly conclude that \( x = y \). This is because the logarithmic functions are strictly increasing (or decreasing) functions, making them one-to-one functions. Remember:

• Both sides of the equation must have logarithms with the same base.
• Once you equate the arguments, you can proceed to solve the resulting equation with standard algebraic techniques.

By understanding this property, you can transform complex logarithmic equations into much simpler linear or other types of equations which are easier to solve.

Solving Linear Equations

Solving linear equations is a foundational skill in algebra. A linear equation in one variable has the form \( ax + b = c \). Solving it involves isolating the variable (often \( x \) or \( n \)) on one side of the equation. The process usually takes a few clear steps:

• First, simplify both sides of the equation if necessary, combining like terms.
• Use addition or subtraction to get all terms with the variable on one side and the constant terms on the other.
• Use multiplication or division to isolate the variable.

In the problem, we started with the equation \(5n - 2 = 8 - 5n\). Adding \(5n\) to both sides helped combine like terms, leading to \(10n - 2 = 8\). Then, adding 2 to both sides isolated the \(10n\) term, resulting in \(10n = 10\). Finally, dividing both sides by 10 gave us \(n = 1\). These steps highlight the efficient tricks of moving terms around to solve for the unknown.

Properties of Logarithms

The properties of logarithms are rules that help us manipulate and solve logarithmic expressions more easily. Knowing these properties will make dealing with logarithmic equations much smoother. Here are some key properties:

• Product Property: \( \log_b(MN) = \log_b(M) + \log_b(N) \)
• Quotient Property: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)
• Power Property: \( \log_b(M^p) = p \cdot \log_b(M) \)
• Change of Base Formula: \( \log_b(M) = \frac{\log_k(M)}{\log_k(b)} \)

These properties allow you to break down complex problems into manageable steps. For example, if tasked with combining or simplifying log expressions, these properties help reorder and simplify the components, making the overall solution path more navigable.