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Chapter 4: Problem 35

Solve logarithmic equation. \(3 x-15=\log _{x} 1 \quad(x>0, x \neq 1)\)

Short Answer

Expert verified
x = 5

Step by step solution

01

Understand the logarithmic equation

The given equation is \(3x - 15 = \log_{x}(1)\). Start by noting that the logarithm of 1 in any base is 0, i.e., \(\log_{b}(1) = 0\).Thus, the equation simplifies to \(3x - 15 = 0\).
02

Solve for x

Now solve the simplified equation \(3x - 15 = 0\). Start by adding 15 to both sides:\(3x = 15\). Then, divide both sides by 3:\(x = 5\).
03

Verify the solution

Verify that the solution satisfies the original conditions: 1. \(x > 0\): \(5 > 0\), which is true.2. \(x eq 1\): \(5 eq 1\), which is also true.Thus, \(x = 5\) is a valid solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Logarithms
Logarithms are the inverse operations to exponentiation. This means that they tell us what power a given base must be raised to in order to achieve a certain number. For example, in the equation \( \log_{2}(8)\) = 3, the number 3 is the power to which 2 must be raised to get 8.

A few important properties of logarithms include:
  • \( \log_{b}(1) = 0\)
  • \( \log_{b}(b) = 1\)
  • \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n)\)
  • \( \log_{b}\left(\frac{m}{n}\right) = \log_{b}(m) - \log_{b}(n)\)
  • \( \log_{b}(m^{n}) = n \log_{b}(m)\)

The equation given in the exercise uses these properties to simplify and solve for the variable x. Thus, understanding these properties can be very helpful.
Solving a Logarithmic Equation
To solve a logarithmic equation involving the variable x, follow these steps:

The provided equation is \(3x - 15 = \log_{x}(1)\). Notice that the right-hand side of the equation simplifies because \( \log_{x}(1) \) is always 0, regardless of the base x. Therefore, the equation becomes \(3x - 15 = 0\).

Next, solve for x by isolating the variable on one side:
  • Start by adding 15 to both sides of the simplified equation: \(3x - 15 + 15 = 0 + 15\) which results in \(3x = 15\).
  • Then, divide both sides by 3 to solve for x: \(x = \frac{15}{3}\) which simplifies to \(x = 5\).

These steps ensure you've correctly simplified and solved the logarithmic equation.
Verification of Solution
After solving for the variable, it is always crucial to verify that the solution is valid and meets all conditions stipulated in the problem.

For this problem, our solution was x = 5. The conditions given were \(x > 0\) and \(x eq 1\).
  • Check the first condition: \(x > 0\). Since 5 is greater than 0, this condition is satisfied.
  • Check the second condition: \(x \eq 1\). Since 5 is not equal to 1, this condition is also satisfied.

Therefore, the solution x = 5 is valid. Verification steps like these are essential as they confirm that the obtained answer is correct and that no mathematical rules were violated in the process.

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Most popular questions from this chapter

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