Logarithm Lab: Compare Linear and Logarithmic Axes
Choose a base and a positive value. Compute its logarithm, check the inverse, and compare equal ratios on linear and logarithmic axes.
Free, with no sign-up.
For High school students and adults reviewing the topic.
About 8 minutes.
How does it work?
- Change a control to update the table and diagram.
- Open a preset to compare settings, then try the challenges.
What will you get?
Compare the model’s outputs, explore its presets, and check two challenges.
What does the model assume?
- Calculations use real logarithms: the base must be positive and different from 1, and the value must be positive. Invalid settings show no numeric result.
- The check uses b raised to x. Change of base divides log₁₀(y) by log₁₀(b). Rounding occurs only in the displayed values.
- Both charts show the same exponential curve and sample points for y from 0.01 to 1000, extended when needed to include the chosen positive value. Horizontal x axes are linear.
- The first vertical axis is linear; the second is logarithmic with one decade per gridline. Equal ratios occupy equal vertical steps on the logarithmic axis.
- The model has no measurement uncertainty and represents a mathematical function, not observed data.
- The base-e preset uses 2.71828 to match the control’s precision. The natural logarithm output uses e itself.
Which setup would you like to try?
What can you change?
Range: 0 to 10 .
Range: -10 to 1000 .
What does the model predict?
| Quantity | Value | Unit |
|---|---|---|
| Domain check | Valid real logarithm | |
| Logarithm x | 3.00000 | |
| Inverse check bˣ | 8.00000 | |
| Common logarithm of y | 0.90309 | |
| Natural logarithm of y | 2.07944 | |
| Common logarithm of b | 0.30103000 | |
| Change-of-base quotient | 3.00000 |
Solid curve: linear vertical scale. Dashed curve: logarithmic vertical scale. Circles mark the same decade values; the square marks the selected value.
For base 2, exponent 3 gives 8. Change of base: 0.90309 divided by 0.30103 gives 3. The logarithmic vertical grid advances by factors of ten.
Can you reach these targets?
Set base 2 and choose the value whose logarithm is 4.
- Target: Base b = 2 ; tolerance 0 .
- Target: Logarithm x = 4 ; tolerance 0.001 .
- Domain check (Valid real logarithm)
Set base 0.5 and choose the value whose logarithm is −2.
- Target: Base b = 0.5 ; tolerance 0 .
- Target: Logarithm x = -2 ; tolerance 0.001 .
- Domain check (Valid real logarithm)