Question

consider the equation \\(\log(3x - 1) = \log_2 8\\). explain why \\(3x - 1\\) is not equal to 8. describe the steps you would take to solve the equation, and state what \\(3x - 1\\) is equal to.

Explanation:

Explain the bases

Using the Logarithmic Evaluation knowledge point

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Evaluate the right side

Using the Logarithmic Evaluation knowledge point

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Rewrite the equation

Using the Solving Logarithmic Equations knowledge point

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Convert to exponential form

Using the Solving Logarithmic Equations knowledge point

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Solve for x

Using the Linear Equations knowledge point

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Answer:

• Why \(3x - 1\) is not equal to 8: The two logarithms have different bases. The left side, \(\log(3x - 1)\), is a common logarithm with an implied base of \(10\), whereas the right side, \(\log_2 8\), has a base of \(2\). You can only equate arguments directly when the bases are identical.
• Steps to solve the equation:

1. Evaluate the right side: \(\log_2 8 = 3\).
2. Rewrite the equation as \(\log(3x - 1) = 3\).
3. Convert the logarithmic equation to its exponential form with base \(10\): \(3x - 1 = 10^3\).
4. Simplify the exponent: \(3x - 1 = 1000\).
5. Solve the linear equation: add \(1\) to both sides to get \(3x = 1001\), then divide by \(3\) to find \(x = \frac{1001}{3}\).

• What \(3x - 1\) is equal to: \(3x - 1 = 1000\)