Problem 31 - Precalculus with Limits

Question

Use the One-to-One Property to solve the equation for \(x .\) \(\log (2 x+1)=\log 15\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 7\)

Step 1: Apply the One-to-One Property of logarithms

Given the equation \(\log (2 x+1) = \log 15\), applying the One-to-One Property of Logarithms (if \(\log_a b = \log_a c\), then \(b = c\)) simplifies the equation to \(2 x+1 = 15\)

Step 2: Simplifying the equation

Next, subtract 1 from both sides to isolate the term with \(x\), resulting in \(2x= 15-1\) which simplifies to \(2x = 14\)

Step 3: Solve for \(x\)

Finally, divide both sides by 2 to solve for \(x\), which gives \(x = 14/2 = 7\)

Key Concepts

Logarithmic Equations

Logarithmic equations are equations that involve logarithms with a variable argument. These equations can often be solved using properties of logarithms and algebraic manipulations. In the problem you are facing, \[\log (2x+1) = \log 15\]you have an example of a simple logarithmic equation. Here, the goal is to find the value of the variable, \(x\).
Understanding how to handle logarithmic equations is crucial since they can appear in various math and real-world applications, like measuring earthquake intensity or calculating compound interest.

• Identify the form of the equation: Check if it's set in a way where the logarithm expressions can be compared directly.
• Apply logarithmic properties: Such as the One-to-One Property to simplify the equation.
• Solve the remaining algebraic equation.

By systematically applying these steps, you can become proficient in solving these types of equations.

Solving Equations

Solving equations is a fundamental skill in mathematics that involves finding the value of unknown variables that make the equation true. With logarithmic equations, the process usually starts with applying relevant properties to simplify the equation.
To solve the equation \[\log (2x + 1) = \log 15\]by using the One-to-One Property, we first equate the arguments of the logarithms:\[2x + 1 = 15\]Next, we solve this simpler algebraic equation by isolating the variable. Here’s how you do it step-by-step:

• Subtract 1 from both sides of the equation to get \[2x = 15 - 1 = 14\]
• Now divide both sides by 2 to isolate \(x\): \[x = \frac{14}{2} = 7\]

This straightforward method follows the basic principles of algebra, where you perform the same operation on both sides of the equation to maintain equality.

Properties of Logarithms

The properties of logarithms are powerful tools to simplify and solve logarithmic equations. Some key properties include the One-to-One Property, product rule, quotient rule, and power rule.
The One-to-One Property is particularly useful when solving an equation like \[\log (2x + 1) = \log 15\]. It states that if\[\log_b A = \log_b B\]then\[A = B\], assuming the logarithm's base and the domain are valid.
Other critical properties are:

• The product rule: \(\log_b(MN) = \log_b M + \log_b N\)
• The quotient rule: \(\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N\)
• The power rule: \(\log_b(M^n) = n\log_b M\)

Understanding and using these properties can help transform complex logarithmic expressions into solvable equations. Applying them strategically makes handling logarithmic equations much more manageable.