Exploring Exponentials & Logarithms — Visual Lab
Exponential growth and its logarithmic inverse are mirror images. Set the base, then watch one curve become the other across the line \(y = x\).
Module 05 · Capstone · Exponentials & Logarithms
This is the Exp & Log Inverse Explorer. One slider sets the base \(b\), and the lab plots
\(y = b^{x}\) and \(y = \log_b x\) together with the mirror line \(y = x\). Drag the tracer point along the
exponential and watch its partner ride the logarithm — the same numbers, with \(x\) and \(y\) swapped.
Module 5 is built on that one idea: a logarithm is an exponential read backwards.
The Inverse Explorer
Drag the base b and the tracer t — or grab the graph itself. Flip on the graphical solver to find where \(b^{x}\) hits a target value.
Inverse Explorer. The solid violet curve is \(y = b^{x}\); the green curve is its inverse \(y = \log_b x\). Each is the other reflected across the dashed \(y = x\) line. The filled point \(P=(t,\,b^{t})\) and its mirror \(P' = (b^{t},\,t)\) prove it: the coordinates simply swap. The exponential's horizontal asymptote \(y = 0\) and the logarithm's vertical asymptote \(x = 0\) are reflections of each other too.
Make It Make Sense
A quick read on what the lab is showing you, and a few experiments to run.
What you're seeing
- The base b sets the shape. When \(b > 1\) the exponential grows; when \(0 < b < 1\) it decays. The logarithm follows.
- The two curves are inverses: \(b^{y} = x \iff y = \log_b x\). Reflect one across \(y = x\) and you land on the other.
- The tracer P and its mirror P′ always have swapped coordinates — that swap is the inverse relationship.
- The exponential never touches \(y = 0\) and the logarithm never touches \(x = 0\); those asymptotes are mirror images.
Try this
- Set b = 2 and drag the tracer. Stop at \(t = 3\): point \(P\) reads \((3, 8)\) and its mirror reads \((8, 3)\) — that says \(2^3 = 8\) and \(\log_2 8 = 3\).
- Slide b below 1 (say \(b = 0.5\)). Watch the banner flip to DECAY and both curves turn over — growth becomes decay.
- Turn on the graphical solver, set \(V = 8\) with \(b = 2\). The vertical line lands on \(x = 3\), because \(\log_2 8 = 3\).
- With the solver on, try \(V = 5\). The answer \(x = \log_2 5 \approx 2.32\) is not a whole number — that is exactly why logarithms exist.
Key Vocabulary & Standards
The words a mathematician uses for exponentials and logarithms — and the TEKS this lab is built to teach.
Exponential Function
A function \(y = b^{x}\) with base \(b > 0,\ b \ne 1\). It grows when \(b > 1\) and decays when \(0 < b < 1\).
Logarithm
The inverse of an exponential: \(y = \log_b x\) answers "to what power must \(b\) be raised to get \(x\)?"
Inverse Functions
Two functions that undo each other. Their graphs are reflections across the line \(y = x\), with \(x\) and \(y\) swapped.
Base
The repeated multiplier \(b\) in \(b^{x}\) and the subscript in \(\log_b x\). The same base ties the inverse pair together.
Asymptote
A line a curve approaches but never reaches. \(y = b^{x}\) hugs \(y = 0\); \(y = \log_b x\) hugs \(x = 0\).
Exponential ↔ Log Form
The conversion at the heart of the unit: \(b^{y} = x \iff y = \log_b x\). Solving one is reading the other.
2A.5 — graphing exponential and logarithmic functions and the inverse relationship between them · 2A.8 — modeling and solving with exponential and logarithmic equations. RDY marks standards frequently assessed as readiness for later courses.