Graphing Logarithmic Functions

Pre-Calculus · Axiom Academy

LESSON Graphing Logarithmic Functions A logarithm is an exponential turned inside out — so its graph is just y=b^x flipped across the line y=x . 1. The Logarithm Is the Exponential, Reflected Saying is the inverse of y=b^x means: whatever point lies on the exponential, the swapped point lies on the logarithm. Swapping coordinates is exactly a reflection across y=x . Watch each blue point on y=2^x fly across the dashed mirror to land on the gold curve — that gold curve is . the defining inverse relationship reflecting a point across y=x swaps its coordinates 2. The Vertical Asymptote and Three Key Points Reflecting flips the exponential's horizontal asymptote y=0 into a vertical asymptote x=0 . So is only defined for , and it plunges toward as x approaches 0 from the right. A tracer slides down the curve toward that wall, then three anchor points light up. x=0 . The domain is ; the curve never crosses to the left of the y -axis. for every base, so every basic log graph passes through (1,0) . , so the curve passes through (b,1) — here (2,1) . The slope , so for the curve rises left to right — slowly, but forever. 3. Domain and Range Trade Places A reflection across y=x swaps the two axes, so it swaps a function's domain and range — and turns a horizontal asymptote into a vertical one. Watch the exponential's anchor point (0,1) and its floor y=0 reflect into the logarithm's anchor (1,0) and its wall x=0 . Passes (0,1) · horizontal asymptote y=0

This is the written version of the interactive lesson above. See the full Pre-Calculus course.