inverse of f(x)=ln(8t),t>0

{ "query": { "display": "inverse $$f\\left(x\\right)=\\ln\\left(8t\\right),\\:t>0$$", "symbolab_question": "FUNCTION#inverse f(x)=\\ln(8t),t>0" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "\\frac{e^{t}}{8}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$\\ln\\left(8t\\right),\\:t>0:{\\quad}\\frac{e^{t}}{8}$$", "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": "x=\\ln\\left(8t\\right)" }, { "type": "interim", "title": "Replace $$t\\:$$with $$x$$", "input": "x=\\ln\\left(8t\\right)", "result": "t=\\ln\\left(8x\\right)", "steps": [ { "type": "step", "primary": "Replace $$t\\:$$with $$x$$", "secondary": [ "Replace $$x\\:$$with $$t$$" ], "result": "t=\\ln\\left(8x\\right)" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZY8+mMcm3KmpETM9SyhE4zAIdcllawnq1qNE+cowJkrTBottbu2aui7TA4izyM3ataN6Hv6MoTMtvtU0IQwXdn8cTGEY1u4y556Dta6zxSzhyYKrRLHk2UoI4WPTb2jaKvvEzkbh/Fj5fvFi0LMLlJk=" } }, { "type": "interim", "title": "Solve for $$x,\\:t=\\ln\\left(8x\\right)$$", "input": "t=\\ln\\left(8x\\right)", "steps": [ { "type": "interim", "title": "Apply log rules", "input": "t=\\ln\\left(8x\\right)", "result": "e^{t}=8x", "steps": [ { "type": "step", "primary": "Use the logarithmic definition: If $$\\log_a\\left(b\\right)=c\\:$$then $$b=a^c$$", "secondary": [ "$$t=\\ln\\left(8x\\right)\\quad\\:\\Rightarrow\\:\\quad\\:e^{t}=8x$$" ], "result": "e^{t}=8x" } ], "meta": { "interimType": "Apply Log Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7V9i85ITPoh3BjTnkCNpNOMA/UsxvvAGebry8C4wy3+HC/itapo7lP0/sRPN25SfrtbNUUDpXEtjwusDD93h4q9ucFazDBJ0idWH6G/mRXo3vbBmbuQNTF0TphKZ8RuvaQrlNmermpNmctl5LmUTDYo2Z+P0nZMR/0+3dTPoQ+vs=" } }, { "type": "interim", "title": "Solve $$e^{t}=8x:{\\quad}x=\\frac{e^{t}}{8}$$", "input": "e^{t}=8x", "result": "x=\\frac{e^{t}}{8}", "steps": [ { "type": "step", "primary": "Switch sides", "result": "8x=e^{t}" }, { "type": "interim", "title": "Divide both sides by $$8$$", "input": "8x=e^{t}", "result": "x=\\frac{e^{t}}{8}", "steps": [ { "type": "step", "primary": "Divide both sides by $$8$$", "result": "\\frac{8x}{8}=\\frac{e^{t}}{8}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{e^{t}}{8}" } ], "meta": { "interimType": "Divide Both Sides Specific 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": "\\frac{e^{t}}{8}" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\left\\{\\ln(8t):t>0\\right\\}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }