inverse of ln(x^3)

{ "query": { "display": "inverse $$\\ln\\left(x^{3}\\right)$$", "symbolab_question": "FUNCTION#inverse \\ln(x^{3})" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "e^{\\frac{x}{3}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$\\ln\\left(x^{3}\\right):{\\quad}e^{\\frac{x}{3}}$$", "steps": [ { "type": "definition", "title": "Function Inverse definition", "text": "A function g is the inverse of function f if for $$y=f\\left(x\\right),\\:\\:x=g\\left(y\\right)\\:$$" }, { "type": "step", "result": "y=\\ln\\left(x^{3}\\right)" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=\\ln\\left(x^{3}\\right)", "result": "x=\\ln\\left(y^{3}\\right)", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=\\ln\\left(y^{3}\\right)" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZTZ65zne7XSU3T2Iz8q9jkL4KqpqHsTkSBgy/axoI6DamTAUhNuvP+oYRx2ZQG6x18Q3tfXX1jSRYh4EFIfu+U5DNOwm/hV0YV0QkO8wKai7EaTcoaJcD/yzWbRAAledhROjRxpO2XYmGPYDdnsqYeQ=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=\\ln\\left(y^{3}\\right)$$", "input": "x=\\ln\\left(y^{3}\\right)", "steps": [ { "type": "interim", "title": "Apply log rules", "input": "x=\\ln\\left(y^{3}\\right)", "result": "y=e^{\\frac{x}{3}}", "steps": [ { "type": "step", "primary": "Apply log rule: $$\\log_{a}\\left(x^b\\right)=b\\cdot\\log_{a}\\left(x\\right)$$", "secondary": [ "$$\\ln\\left(y^{3}\\right)=3\\ln\\left(y\\right)$$" ], "result": "x=3\\ln\\left(y\\right)", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Switch sides", "result": "3\\ln\\left(y\\right)=x" }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3\\ln\\left(y\\right)=x", "result": "\\ln\\left(y\\right)=\\frac{x}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3\\ln\\left(y\\right)}{3}=\\frac{x}{3}" }, { "type": "step", "primary": "Simplify", "result": "\\ln\\left(y\\right)=\\frac{x}{3}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Use the logarithmic definition: If $$\\log_a\\left(b\\right)=c\\:$$then $$b=a^c$$", "secondary": [ "$$\\ln\\left(y\\right)=\\frac{x}{3}\\quad\\:\\Rightarrow\\:\\quad\\:y=e^{\\frac{x}{3}}$$" ], "result": "y=e^{\\frac{x}{3}}" } ], "meta": { "interimType": "Apply Log Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7uEevFnkmsaPDBzpykMhJvKy0+cQg1teFzvbY3+Oda17AOnDcsV4flh1GDqfhbKBXsVBV2q+3sRfeqjWHFfpgTuAxdi2rWaRIC54FE7Ue85kM9jC4hjXgn1UtPAkn2m/J1sD7NfhsPe7eDHrmjY0mE+e2/khsU6lb1o/KZ327eKa9RQa+VFzMcO8uN2y3ZuWk" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "e^{\\frac{x}{3}}" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\ln(x^{3})" }, "showViewLarger": true } }, "meta": { "showVerify": true } }