Problem 98 - College Algebra with Modeling and Visualization

Question

Solve each equation. Approximate answers to four decimal places when appropriate. $$7 \log _{6}(4 x)+5=-2$$

Step-by-Step Solution

Answer

x \approx 0.0417

Step 1: Isolate the logarithmic term

To isolate the logarithmic term, subtract 5 from both sides of the equation:\[ 7 \log_{6}(4x) = -2 - 5 \]This simplifies to:\[ 7 \log_{6}(4x) = -7 \]

Step 2: Divide by coefficient of the logarithmic term

Divide both sides of the equation by 7 to solve for the logarithm:\[ \log_{6}(4x) = \frac{-7}{7} \]This implies:\[ \log_{6}(4x) = -1 \]

Step 3: Convert logarithmic form to exponential form

Convert the logarithmic equation to exponential form to solve for \(4x\):\[ 6^{-1} = 4x \]In exponential form, this is:\[ \frac{1}{6} = 4x \]

Step 4: Solve for x

To find \(x\), divide both sides by 4:\[ x = \frac{1}{6 \times 4} \]This simplifies to:\[ x = \frac{1}{24} \approx 0.0417 \] to four decimal places.

Key Concepts

Exponential Form

The process of converting a logarithmic equation into an exponential form is crucial in solving logarithmic problems. When you have an equation in the form \( \log_b(y) = x \), it can be translated into its exponential form: \( b^x = y \). This conversion is incredibly helpful because it allows you to deal directly with exponential equations, which are often simpler to manipulate and solve.

In the exercise, after isolating the logarithm, we have the equation \( \log_6(4x) = -1 \). In exponential form, this becomes \( 6^{-1} = 4x \). Here, 6 is the base, -1 is the exponent, and \(4x\) is the result of the exponential expression. This step simplifies the process of solving for \( x \) by transforming the equation into a format that is easier to manipulate.

Understanding this concept not only helps in solving logarithmic equations but also enhances the understanding of how logarithms and exponents are related. This connection is essential for both algebraic and real-world applications.

Approximation

Mathematics often requires approximating numbers to a certain level of precision, particularly when dealing with irrational numbers or complex expressions. In this context, once you determine the exact value of \( x \), approximations help you understand or communicate the solution in a practical, everyday context.

• Approximating to four decimal places is common because it provides a balance between precision and simplicity.
• It involves rounding the result to the fourth place after the decimal point, ensuring that calculations remain consistent with expected accuracy levels.

For this equation, we ultimately found \( x = \frac{1}{24} \). Expressed as a decimal, this is approximately \( 0.0417 \) when rounded to four decimal places.

Approximations are not just about rounding numbers but understanding their practical importance in fields like engineering, physics, and everyday problem-solving. Hence, it's critical to master this skill for precision in outputs.

Isolating Variables

The key to solving many equations, including logarithmic ones, is isolating the variable of interest. This process involves manipulating the equation so that the variable sits alone on one side. For logarithmic equations, this often includes moving terms away from the logarithmic expression and simplifying the coefficients.

To solve the given exercise, isolating the logarithmic term \( \log_6(4x) \) was the first step. By subtracting 5 from both sides, the equation became simpler: \( 7\log_6(4x) = -7 \). Following this, the logarithmic expression can be further isolated by dividing both sides by 7, resulting in \( \log_6(4x) = -1 \).

• These steps are crucial for breaking down complex equations into more manageable parts.
• Working systematically to isolate variables makes the problem more straightforward and often reveals links to other solution methods, like conversion to exponential form.

Understanding how to isolate variables is a fundamental skill that applies across various branches of mathematics and serves as a powerful tool for solving equations efficiently.