Problem 62 - Precalculus

Question

Solve the logarithmic equations. Round your answers to three decimal places. $$\log (3 x-5)=-1$$

Step-by-Step Solution

Answer

The solution for \( x \) is approximately 1.700.

Step 1: Understand the Definition of Logarithms

The equation is given in the form \( \log (3x - 5) = -1 \). By definition, the logarithm base 10 of a number equals the power to which 10 must be raised to obtain that number. This means \( 3x - 5 \) needs to equal \( 10^{-1} \).

Step 2: Convert the Logarithmic Equation to its Exponential Form

Rewrite the equation \( \log (3x - 5) = -1 \) in exponential form as \( 3x - 5 = 10^{-1} \). This simplifies to \( 3x - 5 = 0.1 \).

Step 3: Solve for \( x \)

Add 5 to both sides of the equation to isolate the term with \( x \): \[ 3x = 0.1 + 5 \]This simplifies to \( 3x = 5.1 \). Now, divide both sides by 3 to solve for \( x \): \[ x = \frac{5.1}{3} \]Calculate to find \( x \approx 1.700 \).

Step 4: Round the Final Answer

The calculated value for \( x \) is approximately \( 1.700 \) when rounded to three decimal places.

Key Concepts

Understanding Logarithms

Logarithms are powerful tools in mathematics, often used to solve equations involving exponents. At its core, a logarithm answers the question: to what power must a specified base be raised to obtain a certain number? When you see an equation like \( \log (3x - 5) = -1 \), it’s referring to a logarithm with a base of 10. This tells us that 10 raised to the power of -1 will yield \( 3x - 5 \). Understanding how to interpret logarithms is key to solving equations that contain them.

Here are some basics to keep in mind:

• Common logarithms use base 10 and are often written simply as \( \log \).
• The natural logarithm, which uses Euler’s number (\( e \)) as its base, is written as \( \ln \).
• Logarithms are the inverses of exponents; this means that if \( 10^y = x \), then \( \log_{10}x = y \).

Once you grasp the fundamental concept that logarithms convert between exponential and linear representations, solving logarithmic equations becomes much easier.

Converting to Exponential Form

Converting a logarithmic equation into its exponential form is often the critical step to finding a solution. Consider the equation \( \log (3x - 5) = -1 \). This is saying that 10 raised to the power of -1 equals \( 3x - 5 \). To convert this into exponential form, we write it as:

• \( 3x - 5 = 10^{-1} \)

When converted, the equation provides an intuitive way to see what value makes the expression inside the logarithm true. It shifts the problem from a logarithmic perspective to one involving simple arithmetic.

Exponential form can simplify understanding by removing the logarithm, turning the equation into a more familiar arithmetic or algebraic expression. For example, knowing that \( 10^{-1} = 0.1 \) allows us to treat the problem just like any other equation we might solve using basic algebraic techniques.

Solving the Equation

Once the equation is in exponential form, solving it becomes a straightforward task of isolating the variable. From our converted equation \( 3x - 5 = 0.1 \), we need to find \( x \):

• Add 5 to both sides: \( 3x = 0.1 + 5 \)
• This simplifies to \( 3x = 5.1 \)

The next step is to solve for \( x \) by dividing both sides of the equation by 3:

• \( x = \frac{5.1}{3} \)
• Calculating the division gives \( x \approx 1.700 \)

The value of \( x \approx 1.700 \) is the rounded solution to three decimal places. Solving equations in this way involves shifting between forms and applying basic algebraic principles, making it essential to understand each step carefully.