Natural Log Calculator

About Natural Log Calculator

The natural logarithm uses base e ≈ 2.71828. It appears naturally in calculus, growth/decay, and many physical processes.

How to use this calculator

1. Enter a positive number in the input box.
2. Check that the number is greater than 0, because \(\ln(x)\) is only defined for positive \(x\).
3. Click the calculate button to get the natural logarithm.
4. Review the step-by-step result to understand how the value was found.

The formula explained

The formula \(\ln(x)=\log_e(x)\) computes the exponent you place on \(e\) to get \(x\). For example, since \(e^2 \approx 7.389\), we have \(\ln(7.389) \approx 2\).

• x = the positive input number you want to take the natural logarithm of
• e = Euler's number, approximately 2.71828, the base of natural logarithms

Step by step method

1. Start with the number \(x\) you want to evaluate, and make sure \(x>0\).
2. Find the exponent \(y\) such that \(e^y=x\). That exponent is \(\ln(x)\).
3. Use a calculator or known values to estimate the result, then check that the answer makes sense.

Worked example

Problem. Find \(\ln(20)\).

1. We need the exponent \(y\) such that \(e^y=20\).
2. Using a calculator, \(\ln(20) \approx 2.9957\).
3. So raising \(e\) to about 2.9957 gives 20.

Tips and common mistakes

• Only positive numbers have real natural logarithms, so \(\ln(0)\) and \(\ln(-5)\) are not defined in the real numbers.
• If your answer seems too large or too small, remember that \(\ln(1)=0\), \(\ln(e)=1\), and values between 1 and \(e\) give results between 0 and 1.

Frequently asked questions