Documentation/Calc Functions/LN

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LN

Mathematical

Summary:

Calculates the logarithm to the base e or natural logarithm of a positive real number. e is a mathematical constant known as Euler's number, which has a value of approximately 2.71828182845904. The natural logarithm of x is often written as ln(x) or loge(x).

LN(Number)

Returns:

Returns a real number that is the natural logarithm of the specified number.

Arguments:

Number is a positive real number, or a reference to a cell containing that number, that is the value for which the natural logarithm is to be calculated.

• If Number is non-numeric, then LN reports a #VALUE! error.
• If Number is less than or equal to 0, then LN reports an invalid argument error (Err:502).

• For any positive real number x, $\displaystyle{ e^{\text{ln}(x)}=x }$ and $\displaystyle{ \text {ln}(e^x)=x }$.
• A series expansion for the natural logarithm of the positive real number x is:

$\displaystyle{ \text{ln}(x)~=~2 \left(\left(\frac{x-1}{x+1}\right)+\frac{1}{3} \left(\frac{x-1}{x+1}\right)^3 +\frac{1}{5} \left(\frac{x-1}{x+1}\right)^5+...\right) }$

• To convert the natural logarithm of a positive real number x to a common logarithm (base-10), use the formula $\displaystyle{ \text{log}(x)=\text{ln}(x)~/~\text{ln}(10) }$.
• Additional information about natural logarithms can be found at Wikipedia's Natural logarithm page.
• The figure below illustrates the LN function.
LN function

Examples:

Formula Description Returns
=LN(3.5) Here the function returns the natural logarithm of 3.5. 1.25276296849537
=LN(0.5) Here the function returns the natural logarithm of 0.5. −0.693147180559945
=LN(D1) where cell D1 contains the value 1. Here the function returns the natural logarithm of 1. 0
=LN(EXP(1)) Here the function returns the natural logarithm of the constant e. 1
=LN(EXP(345)) Here the function returns the natural logarithm of e raised to the power of 345, which is 345. 345
=EXP(LN(789)) Here the function returns e raised to the power of the natural logarithm of 789, which is 789. 789

LN