solve $\log_{2}x = \log_{5}3 + 1$ by graphing. round to the nearest tenth.
$x \approx$

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Solving Logarithmic Equations", "Solving Exponential Equations Graphically" ], "new_concepts": [], "current_concepts": [ "Solving Logarithmic Equations", "Solving Exponential Equations Graphically" ] } ## Set up the system of equations Using the *Solving Exponential Equations Graphically* knowledge point $$ \begin{aligned} &y_1 = \log_2(x) \ &y_2 = \log_5(3) + 1 \end{aligned} $$ ## Evaluate the constant term Using the *Solving Logarithmic Equations* knowledge point $$ \begin{aligned} &\log_5(3) = \frac{\ln(3)}{\ln(5)} \approx 0.6826 \ &y_2 \approx 0.6826 + 1 = 1.6826 \end{aligned} $$ ## Solve for x algebraically Using the *Solving Logarithmic Equations* knowledge point $$ \begin{aligned} &\log_2(x) = \log_5(3) + 1 \ &x = 2^{\log_5(3) + 1} \ &x = 2^{1.6826} \approx 3.2099 \end{aligned} $$ ## Round to the nearest tenth Using the *Solving Logarithmic Equations* knowledge point $$ x \approx 3.2 $$ Solve $\log_2x = \log_53 + 1$ by graphing. Round to the nearest tenth. $x \approx$ 3.2 { "elements": [ { "type": "functiongraph", "params": [ { "js": "Math.log(x)/Math.log(2)", "latex": "\log_2(x)" }, 0.01, 10 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "name": "y = \log_2(x)", "withLabel": true } }, { "type": "functiongraph", "params": [ { "js": "Math.log(3)/Math.log(5) + 1", "latex": "\log_5(3) + 1" }, -10, 10 ], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2, "name": "y = \log_5(3) + 1", "withLabel": true } }, { "type": "point", "params": [ [3.21, 1.68] ], "properties": { "name": "Intersection (3.2, 1.7)", "size": 4, "color": "#5583F2", "withLabel": true } } ], "timestamps": [0.5, 1.0] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Solving Logarithmic Equations" ] }

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Question

Explanation:

Set up the system of equations

Evaluate the constant term

Solve for x algebraically

Round to the nearest tenth

Answer: