Question

Find the value of $\ln 10$.

Answer 1

Hint: The natural logarithm of a number is the power to which $e$ would have to be raised to equal that number. Here the number given is $10$. So we have to find the value of $n$ such that ${e^n}$ is $10$.

Complete step-by-step answer:
Given the number \[10\].
We are asked to find $\ln 10$.
Here $\ln $ denotes natural logarithm.
The natural logarithm of a number is the power to which $e$ would have to be raised to equal that number, where $e$ is the exponential constant.
So we have to find the value of $n$ such that ${e^n}$ is $10$.
We know $e = 2.71828$
This gives ${e^n} = {(2.71828)^n}$
Now ${e^n} = 10 \Rightarrow {(2.71828)^n} = 10$
Therefore, taking $n^{th}$ root on both sides we get,
\[2.71828 = {10^{\dfrac{1}{n}}}\]
Thus we can see $n = 2.302$.

$\therefore \ln 10 = 2.302$

Additional Information: The natural logarithm can be defined for any positive real number. Logarithms can be defined for any positive base other than one. For instance, the base-$2$ logarithms are also called binary logarithms.
Exponential function is the inverse of logarithmic function. Both logarithmic functions and exponential functions are widely used in Mathematics as well as in other subjects like Physics.

Note: The natural logarithm is also denoted as ${\log _e}x$. The natural logarithm of $e$ itself, $\ln e = 1$, because ${e^1} = e$ while the natural logarithm of one is zero, since ${e^0} = 1$.
If the base is different from the exponential constant it will be specified in the question. Other common base used in Mathematics for logarithm is ten. Anyway logarithm to any base means the power to which the base would have to be raised to equal that number.