Question

How do you solve $ \ln x = - 7? $

Answer 1

Hint:Let us understand in a general way, the function of a logarithm, Consider a logarithm of $ a $ with base $ b $ equal to $ c $ , it means that when $ c $ is raised to the power of $ b $ then it will equal to $ a $ , let’s understand this in mathematical way or in equations $ \Rightarrow {\log _b}a = c $ It is the mathematical form of the first part of the consideration. Now coming to the second part $ \Rightarrow {b^c} = a $ It is the mathematical form of the second part. Hope that you got to understand the logarithmic function. $ \ln $ is a special logarithmic function which has a fixed base equals $ e $ Try to write the problem in equation form and then solve it further.

Complete step by step solution:
We have to find the value of $ x $ using some properties logarithm and
exponential, Now let’s come to the question, we have $ \ln x = - 7 $ , we can also write this in $ \log $ form as
 $ {\log _e}x = - 7 $ _____(I)

Logarithmic function has a property that if it is raised on the power of its base then only argument of the logarithm left, let us understand this with an example $ {\log _a}b $ , if we raise $ {\log _a}b $ in the power of $ a $ then only $ b $ will left
 $ \Rightarrow {a^{{{\log }_a}b}} = b $

Now we will raise both sides of equation (I) on the power of $ e $ , we will get
 $
\Rightarrow {e^{{{\log }_e}x}} = {e^{ - 7}} \\
\Rightarrow x = {e^{ - 7}} \\
 $

We got the required solution for $ \ln x = - 7 $ which is $ x = {e^{ - 7}} $
If you want numerical value then calculate this on a calculator you will get a value close to $ x = 9.12
\times {10^{ - 4}} $

Note: The functions $ y = {e^x}\;\& \;y = \ln x $ are inverse functions of each other. Base of a logarithm function should be greater than zero and can’t be equal to one.
Argument of the logarithm function is always positive.