inverse of y=ln(x-1)-ln(2)-1

{ "query": { "display": "inverse $$y=\\ln\\left(x-1\\right)-\\ln\\left(2\\right)-1$$", "symbolab_question": "FUNCTION#inverse y=\\ln(x-1)-\\ln(2)-1" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "2e^{x+1}+1", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$\\ln\\left(x-1\\right)-\\ln\\left(2\\right)-1:{\\quad}2e^{x+1}+1$$", "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-1\\right)-\\ln\\left(2\\right)-1" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=\\ln\\left(x-1\\right)-\\ln\\left(2\\right)-1", "result": "x=\\ln\\left(y-1\\right)-\\ln\\left(2\\right)-1", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=\\ln\\left(y-1\\right)-\\ln\\left(2\\right)-1" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZRIWrAVj6y0+Bw3xvyQjAorhbvtSXsslnaqjm4gUebvt+mYo+NX/AKmHymeOa3NnrQI8+hkLyApZ1/OS023CTb/WwPs1+Gw97t4MeuaNjSYTcMbpTwO1qcz1V3htDWv+f/MC5Yutg6fcuEuopj//+yc=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=\\ln\\left(y-1\\right)-\\ln\\left(2\\right)-1$$", "input": "x=\\ln\\left(y-1\\right)-\\ln\\left(2\\right)-1", "steps": [ { "type": "step", "primary": "Switch sides", "result": "\\ln\\left(y-1\\right)-\\ln\\left(2\\right)-1=x" }, { "type": "interim", "title": "Move $$\\ln\\left(2\\right)\\:$$to the right side", "input": "\\ln\\left(y-1\\right)-\\ln\\left(2\\right)-1=x", "result": "\\ln\\left(y-1\\right)-1=x+\\ln\\left(2\\right)", "steps": [ { "type": "step", "primary": "Add $$\\ln\\left(2\\right)$$ to both sides", "result": "\\ln\\left(y-1\\right)-\\ln\\left(2\\right)-1+\\ln\\left(2\\right)=x+\\ln\\left(2\\right)" }, { "type": "step", "primary": "Simplify", "result": "\\ln\\left(y-1\\right)-1=x+\\ln\\left(2\\right)" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "\\ln\\left(y-1\\right)-1=x+\\ln\\left(2\\right)", "result": "\\ln\\left(y-1\\right)=x+\\ln\\left(2\\right)+1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "\\ln\\left(y-1\\right)-1+1=x+\\ln\\left(2\\right)+1" }, { "type": "step", "primary": "Simplify", "result": "\\ln\\left(y-1\\right)=x+\\ln\\left(2\\right)+1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Apply log rules", "input": "\\ln\\left(y-1\\right)=x+\\ln\\left(2\\right)+1", "result": "y-1=2e^{x+1}", "steps": [ { "type": "step", "primary": "Use the logarithmic definition: If $$\\log_a\\left(b\\right)=c\\:$$then $$b=a^c$$", "secondary": [ "$$\\ln\\left(y-1\\right)=x+\\ln\\left(2\\right)+1\\quad\\:\\Rightarrow\\:\\quad\\:y-1=e^{x+\\ln\\left(2\\right)+1}$$" ], "result": "y-1=e^{x+\\ln\\left(2\\right)+1}" }, { "type": "interim", "title": "Expand $$e^{x+\\ln\\left(2\\right)+1}:{\\quad}2e^{x+1}$$", "input": "e^{x+\\ln\\left(2\\right)+1}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "result": "=e^{\\ln\\left(2\\right)}e^{x+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Simplify $$e^{\\ln\\left(2\\right)}:{\\quad}2$$", "input": "e^{\\ln\\left(2\\right)}", "result": "=2e^{x+1}", "steps": [ { "type": "step", "primary": "Apply log rule: $$a^{\\log_{a}\\left(b\\right)}=b$$", "result": "=2", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YI5QbaXdDiUthjB9nLIneMXxNHg0rONgTaEu1U9A2zCbwLLf/4qFQM8wXvPHcutMOsi0ZMrmx6dIX+R0AW4q1EUqTd96MWTKI6Kr2Ib0iQDan2z/3kXxOb0ZWofMdlwE/bw41uva7vfx7Tp0nwU+uA==" } }, { "type": "step", "result": "y-1=2e^{x+1}" } ], "meta": { "interimType": "Apply Log Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vnALIVV4XNQt/VVBzebWLWQJ721b7Vgq28wbEvB6s4s52FciCV6Q/ZuTzBHIPdDywh1dx4QxVQTFFuRGwejJdxBZfaSS8J7oFLak19JT8wPJxKCFhUvEheg+qeRFqfIngQUxJPyUNnGfVirkcwpVOz4+ruY5kuligXlC0CtPrr/nGcX57Hy3fYoom/ZxpK5G" } }, { "type": "interim", "title": "Solve $$y-1=2e^{x+1}:{\\quad}y=2e^{x+1}+1$$", "input": "y-1=2e^{x+1}", "result": "y=2e^{x+1}+1", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "y-1=2e^{x+1}", "result": "y=2e^{x+1}+1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "y-1+1=2e^{x+1}+1" }, { "type": "step", "primary": "Simplify", "result": "y=2e^{x+1}+1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "2e^{x+1}+1" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\ln(x-1)-\\ln(2)-1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }