Solve each equation. 3x - 15 = log_x 1 (x>0, x≠1)

Verified step by step guidance

1. Identify the equation given: \(3x - 15 = \log_{x} 1\) with the conditions \(x > 0\) and \(x \neq 1\).
2. Recall the logarithm property that \(\log_{a} 1 = 0\) for any base \(a > 0\) and \(a \neq 1\). Therefore, replace \(\log_{x} 1\) with 0 in the equation.
3. Rewrite the equation as \(3x - 15 = 0\) after substituting the logarithm value.
4. Solve the linear equation \(3x - 15 = 0\) by isolating \(x\): add 15 to both sides to get \(3x = 15\), then divide both sides by 3 to find \(x\).
5. Check the solution against the domain restrictions \(x > 0\) and \(x \neq 1\) to ensure it is valid.

Key Concepts

Properties of Logarithms

Logarithms are the inverse operations of exponentiation. Understanding their properties, such as the domain restrictions (x > 0 and base ≠ 1) and how to manipulate logarithmic expressions, is essential for solving equations involving logs.

Solving Linear Equations

Linear equations involve variables raised to the first power and can be solved using basic algebraic operations like addition, subtraction, multiplication, and division. Recognizing and isolating the variable is key to finding solutions.

Domain Restrictions in Logarithmic Equations

Logarithmic functions have domain restrictions: the argument must be positive, and the base cannot be 1. These restrictions limit the possible solutions and must be considered to ensure valid answers.