Problem 76 - Intermediate Algebra

Question

Solve each equation. $$ \log 3 x=\log 9 $$

Step-by-Step Solution

Answer

The solution is \( x = 3 \).

Step 1: Understand the Problem

The given equation is \( \log(3x) = \log(9) \). This means the logarithm of \(3x\) is equal to the logarithm of 9.

Step 2: Use the Property of Logarithms

When you have \( \log(a) = \log(b) \), you can conclude that \( a = b \). Apply this property to our equation: \( 3x = 9 \).

Step 3: Solve for x

Now we solve for \( x \) in the equation \( 3x = 9 \). To isolate \( x \), divide both sides by 3: \[ x = \frac{9}{3} \] This simplifies to \( x = 3 \).

Key Concepts

Properties of Logarithms

When working with logarithms, it's vital to understand their properties, as they help in simplifying and solving logarithmic equations. One essential property is that if the logarithms of two expressions are equal, then the expressions themselves must be equal as well. Specifically, if \( \log(a) = \log(b) \), it logically follows that \( a = b \). This property is frequently used to transition from dealing with logarithms to solving a more straightforward algebraic equation.

Another critical set of properties involves how logarithms deal with multiplication, division, and powers:

• **Product Rule**: \( \log_b(MN) = \log_b(M) + \log_b(N) \)
• **Quotient Rule**: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)
• **Power Rule**: \( \log_b(M^p) = p \cdot \log_b(M) \)

These rules can significantly simplify complex logarithmic expressions and are often the first step in untangling logarithmic equations.

Equation Solving

Solving equations requires a systematic approach that involves several steps. In the context of log equations like \( \log(3x) = \log(9) \), the first step is to recognize the property of logarithms which allows you to equate the insides of the logs once the logs themselves are equal.

With \( \log(3x) = \log(9) \), the next logical move is using the property \( \log(a) = \log(b) \Rightarrow a = b \) to deduce that \( 3x = 9 \). With the insights from the property, the equation solving moves from solving a logarithmic equation to a simple algebraic equation.

• Divide each side by 3 to isolate the variable \( x \): \( x = \frac{9}{3} \)
• This results in \( x = 3 \)

Recognizing key properties of logarithms strengthens the ability to solve these equations efficiently, translating the logarithmic problem into a straightforward algebraic equation seeking a solution.

Intermediate Algebra

In intermediate algebra, students often encounter logarithmic equations, an intersection of algebra and pre-calculus. Understanding how to manipulate and solve these equations is crucial for building a stronger foundation in mathematics.

Concepts like solving \( 3x = 9 \) after simplifying \( \log(3x) = \log(9) \) showcase an intermediate algebra skill — balancing and simplifying equations. Such a phase of equation solving incorporates:

• Applying arithmetic to isolate variables after transforming log equations
• Evaluating expressions and understanding equality \( a = b \)

These actions aren’t just procedural; they help transition skills essential in higher math. Solving \( x \) ensures comprehension of value and variable manipulation, pivotal for progressing further in algebra and beyond into more advanced mathematical realms.