Log vs Ln: Understanding the Key Differences

Differences between Log and Ln come down to the base they use and how we read them. Log (short for logarithm) usually means logarithm with base 10 in school problems, while Ln means logarithm with base e (where e ≈ 2.718), a special number that shows up a lot in nature and growth. Think of logs like “questions about how many times to multiply a number to get another number.” For example, log base 10 of 100 is 2 because 10 × 10 = 100. Ln asks the same kind of question but with e instead of 10. But the real difference runs much deeper than just the base it shows up in their graphs, their derivatives and their applications.

Table of Contents

• Difference Between Log and Ln
• Where to Use Log and Ln
• The Relationship Between Log and Ln

Difference Between Log and Ln

Although Log and Ln are both logarithmic functions, they differ in the base they use. Understanding the difference between Log and Ln is essential for solving equations, simplifying expressions, and working with exponential functions.

Where to Use Log and Ln

One of the biggest sources of confusion is that students treat log and ln as interchangeable without realising that the two functions belong to genuinely different mathematical worlds.

Log (base 10) is the go-to for:

• pH in chemistry: pH = −log[H⁺], where [H⁺] is the hydrogen ion concentration. Every unit drop in pH means a 10× increase in acidity because the scale is base 10.

• Richter scale (earthquakes): A magnitude 7 quake is 10× more powerful than a magnitude 6. Again, base 10 scaling.

• Decibel scale (sound intensity): Sound levels in dB use log base 10 of the intensity ratio.

• Engineering and signal processing: Gain, attenuation, and frequency response are all expressed in log base 10.

• Your calculator's ‘log’ button always gives base 10 by default.
  
  

Ln (base e) is the go-to for:

• Calculus: The derivative of ln(x) is simply 1/x. This is why calculus textbooks default to ln almost exclusively.

• Exponential growth and decay: Population growth, radioactive decay, and cooling all follow the form N = N₀·eᵏᵗ, which means ln is the natural tool for solving them.

• Continuous compound interest: The formula A = Peʳᵗ uses e as its base, ln undoes it.

• Thermodynamics and entropy: Many physics formulas are written with ln because the underlying mathematics involves e.

• Statistics: Log-likelihood functions and the normal distribution both use ln.

• Biology: Bacterial growth curves, enzyme kinetics (Michaelis-Menten), and pharmacokinetics all involve ln.

Rule of thumb: if the problem involves a measurement scale designed by humans (pH, decibels, Richter), use log. If it involves something that happens in nature on its own (growth, decay, compounding), use ln.

The Relationship Between Log and Ln

Log and ln are the same type of function with different bases, and you can always convert one to the other.

• The Change of Base Formula

log_b(x) = log_a(x) / log_a(b)

Applying this to convert between log and ln:

• Converting log → ln:

log(x) = ln(x) / ln(10)

Since ln(10) ≈ 2.303: ln(x) = 2.303 × log(x)

• Converting ln → log:

ln(x) = log(x) / log(e)

Since log(e) ≈ 0.4343: log(x) = 0.4343 × ln(x)

Why the Factor 2.303?

Since e ≈ 2.718 is smaller than 10, the natural logarithm of any number x is always larger than its common logarithm (for x > 1). You need to raise e to a higher power to reach the same number x that you would reach by raising 10 to a smaller power. That ratio is ln(10) ≈ 2.303 it is the ‘stretching factor’ between the two scales.

Example: In chemistry, the Nernst equation is written with ln. To switch it to log base 10 (easier for quick mental calculations), every instance of ln gets multiplied by 2.303:

E = E° − (RT/nF) · ln(Q) becomes E = E° − (2.303 RT/nF) · log(Q)