Logarithms
Natural Logarithm vs Common Logarithm
The natural logarithm ln(x) uses base e. The common logarithm log_10(x) uses base 10. They measure the same input on different exponent scales.
Reviewed
ln and log10 side by side
They are the same operation with different bases, so their laws match while their numerical values differ.
| Notation | Base | Exact anchor values | Common context |
|---|---|---|---|
| ln(x) | e | ln(1) = 0, ln(e) = 1 | Continuous growth, calculus, differential equations |
| log(x) or log10(x) | 10 | log10(1) = 0, log10(10) = 1 | Orders of magnitude, decibels, pH calculations |
Because some calculators and programming languages use log to mean ln, always check the stated base.
What the two bases mean
ln(x) asks which power of e equals x. The constant e is approximately 2.71828 and arises naturally in continuous growth and calculus.
log_10(x) asks which power of 10 equals x. It aligns with decimal place value, making powers of ten especially easy to interpret.
ln(x)=log_e(x); common log=log_10(x)Worked example 1
Compare ln(100) and log_10(100)
The input is the same but the exponent scales differ.
log_10(100)=2 because 10^2=100ln(100) is the exponent y with e^y=100
Result: log_10(100)=2; ln(100) is approximately 4.60517
Same laws, different scale
Both functions obey the product, quotient, and power rules because those rules come from exponent laws. Their numerical outputs differ by a constant conversion factor.
The base-change formula lets either one compute a logarithm in any valid base.
log_10(x)=ln(x)/ln(10); ln(x)=log_10(x)/log_10(e)Pick the base from the model
- Natural log: continuous growth, decay, calculus, and compound processes.
- Common log: orders of magnitude and powers-of-ten scales.
- Either base: solving exponent equations through a quotient of logarithms.
Worked example 2
Solve e^(0.4t)=7
Natural log directly reverses the exponential base.
ln(e^(0.4t))=ln(7)0.4t=ln(7)t=ln(7)/0.4
Result: t is approximately 4.86478
Choosing a logarithm
Use ln when working with continuous growth, derivatives, exponential decay, or formulas involving e. Use common log when a problem is built around powers of ten or a decimal logarithmic scale.
For solving a single exponential equation, any consistent base gives the same final solution after division.
Questions about natural log vs common log
Is ln just another notation for log?
It specifically means base e. Plain log notation varies by context.
Will different log bases give different solutions to an exponential equation?
Intermediate values differ, but consistent use of the change-of-base formula gives the same solution.