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Natural Logarithm of 10 — Definition, Formula & Examples

The natural logarithm of 10, written ln⁡10\ln 10, is the power you must raise ee to in order to get 10. Its approximate value is 2.302585.

ln⁡10\ln 10 is the unique real number xx satisfying ex=10e^x = 10, where e≈2.71828e \approx 2.71828 is Euler's number. Its value to six decimal places is 2.3025852.302585.

Key Formula

ln⁡10=log⁡e10≈2.302585\ln 10 = \log_e 10 \approx 2.302585
Where:
  • ln⁡\ln = The natural logarithm function (base e)
  • ee = Euler's number, approximately 2.71828

How It Works

Because e2.302585≈10e^{2.302585} \approx 10, the value ln⁡10\ln 10 acts as a conversion factor between natural logarithms and common (base-10) logarithms. The change of base formula states log⁡10x=ln⁡xln⁡10\log_{10} x = \frac{\ln x}{\ln 10}, so dividing any natural logarithm by ln⁡10\ln 10 converts it to a base-10 logarithm. This constant appears frequently when solving exponential equations that mix bases ee and 1010.

Worked Example

Problem: Convert ln⁡50\ln 50 to a common logarithm (base 10) using ln⁡10\ln 10.
Apply the change of base formula: Divide the natural logarithm by ln⁡10\ln 10 to convert to base 10.
log⁡1050=ln⁡50ln⁡10\log_{10} 50 = \frac{\ln 50}{\ln 10}
Substitute known values: A calculator gives ln⁡50≈3.91202\ln 50 \approx 3.91202. Use ln⁡10≈2.30259\ln 10 \approx 2.30259.
log⁡1050≈3.912022.30259≈1.69897\log_{10} 50 \approx \frac{3.91202}{2.30259} \approx 1.69897
Answer: log⁡1050≈1.699\log_{10} 50 \approx 1.699, which matches the common logarithm of 50.

Why It Matters

Whenever you solve exponential growth or decay problems using ee but need a base-10 answer (for example, pH in chemistry or decibels in physics), ln⁡10\ln 10 is the bridge between the two logarithm bases. Memorizing that ln⁡10≈2.303\ln 10 \approx 2.303 saves time on exams where calculator access is limited.

Common Mistakes

Mistake: Confusing ln⁡10\ln 10 with log⁡10e\log_{10} e.
Correction: These are reciprocals: log⁡10e≈0.4343\log_{10} e \approx 0.4343 while ln⁡10≈2.3026\ln 10 \approx 2.3026. Notice ln⁡10×log⁡10e=1\ln 10 \times \log_{10} e = 1.

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