Solve exponential equations with tables, graphs, and inverse steps. Compare growth and decay models quickly. Export clean results for class, homework, study, or review.
Use the general or natural model. Then review the function, inverse result, derivatives, integral, table, and graph.
Sample model: y = 3 × 2x + 1
| x | 2x | y = 3 × 2x + 1 |
|---|---|---|
| -2 | 0.25 | 1.75 |
| -1 | 0.5 | 2.5 |
| 0 | 1 | 4 |
| 1 | 2 | 7 |
| 2 | 4 | 13 |
| 3 | 8 | 25 |
y = a × b(cx + d) + k
Derivative: y′ = a × c × ln(b) × b(cx + d)
Second derivative: y′′ = a × (c × ln(b))2 × b(cx + d)
Inverse: x = (logb((y - k) / a) - d) / c
y = a × e(rx + d) + k
Derivative: y′ = a × r × e(rx + d)
Second derivative: y′′ = a × r2 × e(rx + d)
Inverse: x = (ln((y - k) / a) - d) / r
Unit growth factor for the general form is bc.
Unit growth factor for the natural form is er.
Percent change per unit x = (unit factor - 1) × 100.
Exponential functions appear in algebra, finance, biology, and physics. They model repeated multiplication. That makes them different from linear functions, which change by equal amounts. This calculator helps you inspect both behaviors quickly. You can evaluate a single point, generate a full table, and visualize the curve on one page.
The coefficient a stretches or flips the curve. The base b controls how fast the general model changes. The rate r does the same for the natural model. The multiplier c changes horizontal speed. The shift d moves the exponent expression. The value k moves the graph up or down. These inputs let you test many classroom and real-world cases.
When the unit factor is greater than one, the function shows growth. When it stays between zero and one, the function shows decay. That idea is important in compound interest, population change, radioactive loss, and cooling models. The calculator converts that factor into a percentage change per unit x, which is easier to interpret.
The derivative tells you how fast the function changes at a chosen point. The second derivative shows curvature. The definite integral adds total accumulation across an interval. These values help in advanced math work, including optimization, modeling, and rate interpretation. Students can compare direct evaluation with calculus results in the same view.
A function value alone can hide the bigger pattern. A table shows repeated change. A graph shows shape, steepness, and shift. Together they make exponential behavior easier to understand. The CSV and PDF options also help with assignments, notes, and reports.
It evaluates exponential functions, estimates inverse x from a target y, computes derivatives, second derivatives, definite integrals, and creates a graph with a table.
The general model uses a custom base b. The natural model uses e as the base. Both describe exponential change, but the natural form is common in calculus.
It is growth when the unit factor is greater than 1. In the general form, that factor is b^c. In the natural form, it is e^r.
Inverse x can fail when the function is constant or when the logarithm input becomes zero or negative. The calculator explains that case in the result.
The derivative measures the instant rate of change. It tells you how quickly the output is increasing or decreasing at the selected x value.
The range builds the table and graph. It lets you study the full trend instead of seeing only one computed point.
No. This calculator restricts the general model to positive bases that are not equal to 1. That keeps real-valued exponential rules valid.
The CSV option saves the generated table for spreadsheets. The PDF option saves the summary and table in a portable document format.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.