Question

How do you evaluate $\ln 4 + \ln 7$ ?

Answer 1

Hint: In order to evaluate this, we use the logarithmic properties. We know that $\ln \left( {m \times n} \right) = \ln m + \ln n$. Therefore, accordingly $\ln x = \ln 4 + \ln 7$ can be written as $\ln x = \ln \left( {4 \times 7} \right)$. On simply solving this further, we can find our answer.

Complete step by step solution:
In this question, we are asked to evaluate the value of $ln4 + ln7$.
\[\ln \] refers to the natural logarithm, which simply means that the logarithm has the base as exponent $e$
Let $\ln 4 + \ln 7$ be equal to $ x$
Therefore we have: $ x = \ln 4 + \ln 7$
Now according to the properties of logarithm, $\ln \left( {m \times n} \right) = \ln m + \ln n$
Therefore, $ x = \ln \left( {4 \times 7} \right)$
$ \Rightarrow x = \ln 28$

Therefore the value of the given logarithm is $\ln 28$.

Note: \[\ln \] refers to natural logarithm. It can be referred to as the power which the base ‘e’ has to be raised to in order to obtain a number called its log number. Thus it is expressed as ${\log _e} = x$ , where x is any number. ‘e’ refers to the exponential function whose value is roughly equal to $2.718281$. Now, the difference between log and \[\ln \] is the fact that log is defined for the base $10$ while \[\ln \] is defined for the base ‘e’.
Some common properties of natural log are:
Product rule:
$\ln \left( x \right)\left( y \right) = \ln x + \ln y$ → the natural log of the product of $x$ and $y$ is the sum of the two numbers.
Quotient rule:
$\ln \left( {\dfrac{x}{y}} \right) = \ln x - \ln y$ → the natural log of the division of $x$ and $y$ is the difference of the two numbers.
Reciprocal rule:
$\ln \left( {\dfrac{1}{x}} \right) = - \ln x$ → the natural log of the reciprocal of $x$ is the opposite of \[\ln \] of $x$
Power rule:
$\ln \left( {{x^y}} \right) = y\ln \left( x \right)$ → the natural log of $x$ raised to the power of $y$ is $y$ times the $\ln \left( x \right)$