Problem

For \(E(x)~=~2^{x}\) with points \((0,~1)\), \((1,~2)\), and \((2,~4)\), determine its logarithmic inverse. State the inverse equation, reversed points, domain, range, vertical asymptote, and behavior.

Actual exponential source graph without its inverse curve.

Big Picture

What this problem is really about

Think of an inverse as the same relationship read backward: every input-output pair trades places. We’ll use that swap both algebraically and graphically, reflecting points across the line y equals x. That single reflection also explains why domain and range exchange roles and why a horizontal asymptote becomes vertical.

Answer choices

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Preview only

inverse equation = L(x)=2ˣ; reversed points = (0,1),(1,2),(2,4); domain = (-∞,∞); range = (0,∞); vertical asymptote = none; behavior = increasing

inverse equation = L(x)=log₂(x); reversed points = (1,0),(2,1),(4,2); domain = (0,∞); range = (-∞,∞); vertical asymptote = x=0; behavior = increasing

inverse equation = L(x)=logₓ(2); reversed points = (1,0),(2,1),(4,2); domain = (-∞,∞); range = (0,∞); vertical asymptote = y=0; behavior = increasing

inverse equation = L(x)=log₂(x); reversed points = (0,1),(1,2),(2,4); domain = (-∞,∞); range = (0,∞); vertical asymptote = y=0; behavior = increasing

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For \(E(x)~=~2^{x}\) with points \((0,~1)\), \((1,~2)\), and \((2,~4)\), determine its logarithmic inverse. State the inverse equation, reversed points, domain, range, vertical asymptote, and behavior.

What this problem is really about

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