Question
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Solve for \( x \) \[ \log _{3}(-4 x+10)=3 \]

Ask by Bradley Reeves. in the United States
Mar 14,2025

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Tutor-Verified Answer

Answer

\( x = -4.25 \)

Solution

Solve the equation \( \log_{3}(-4x+10)=3 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\log_{3}{\left(-4x+10\right)}=3\) - step1: Find the domain: \(\log_{3}{\left(-4x+10\right)}=3,x<\frac{5}{2}\) - step2: Convert the logarithm into exponential form: \(-4x+10=3^{3}\) - step3: Evaluate the power: \(-4x+10=27\) - step4: Move the constant to the right side: \(-4x=27-10\) - step5: Subtract the numbers: \(-4x=17\) - step6: Change the signs: \(4x=-17\) - step7: Divide both sides: \(\frac{4x}{4}=\frac{-17}{4}\) - step8: Divide the numbers: \(x=-\frac{17}{4}\) - step9: Check if the solution is in the defined range: \(x=-\frac{17}{4},x<\frac{5}{2}\) - step10: Find the intersection: \(x=-\frac{17}{4}\) The solution to the equation \( \log_{3}(-4x+10)=3 \) is \( x = -\frac{17}{4} \) or \( x = -4.25 \).

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The Deep Dive

To solve the equation \( \log _{3}(-4 x + 10) = 3 \), we first rewrite the logarithmic equation in its exponential form. This means we express it as: \[ -4x + 10 = 3^3 \] Calculating \( 3^3 \), we find: \[ 3^3 = 27 \] Now our equation looks like this: \[ -4x + 10 = 27 \] Next, we will isolate \( x \). Start by subtracting 10 from both sides: \[ -4x = 27 - 10 \] \[ -4x = 17 \] Now, to solve for \( x \), divide both sides by -4: \[ x = \frac{17}{-4} \] \[ x = -\frac{17}{4} \] Finally, it's important to check if this solution is valid by substituting it back into the original logarithmic expression. The original expression is: \[ \log_{3}(-4(-\frac{17}{4}) + 10) = \log_{3}(17 + 10) = \log_{3}(27) \] And since \( \log_{3}(27) = 3 \), it confirms that our solution is correct. Thus, the final answer is: \[ \boxed{-\frac{17}{4}} \]

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