Question

How do you solve $\ln (x - 5) = 3?$

Answer 1

The equation consists of the natural logarithm, convert it into exponential form and then further solve the equation. Logarithmic functions can be converted into exponential form as follows
${\log _a}b = c \Rightarrow {a^c} = b$ , where “a” and “b” are base and argument of the logarithmic function respectively whereas “c” is the exponent or power or the result of the logarithm function.

Complete step by step solution:
For solving an equation which has logarithm function in it, we have to first convert the logarithmic form into exponential form, to make the equation less complex and simple.
In order to convert logarithm function, we have to know about the relation between logarithm and exponential function which is given as follows
${\log _a}b = c \Rightarrow {a^c} = b$ , where “a” and “b” are the base and argument of the logarithmic function respectively.
So now converting the given logarithmic equation into exponential form
$\Rightarrow \ln (x - 5) = 3$
We can also write it as
$\Rightarrow {\log _e}(x - 5) = 3 \\\ \Rightarrow x - 5 = {e^3} \\\ \Rightarrow x = {e^3} + 5 \\\$
So $x = {e^3} + 5$ is the required solution for the logarithmic equation $\ln (x - 5) = 3$
You can leave the answer as it is, because $e$ is an irrational number so if we calculate the value of $x = {e^3} + 5$ numerically then it will come as $25.08553692...$

Note: “$\ln$” denotes the natural logarithm function, which has a base equals to $e$ .Values of logarithm can be positive or negative but logarithm never takes a negative argument, that is its arguments should always be greater than zero and also its base should be greater than zero and not equals to one. These are some necessary rules of logarithm function for taking argument and base.