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Question

Question: How do you solve \( \ln x = - 7? \)...

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

Explanation

Solution

Let us understand in a general way, the function of a logarithm, Consider a logarithm of aa with base bb equal to cc , it means that when cc is raised to the power of bb then it will equal to aa , let’s understand this in mathematical way or in equations logba=c\Rightarrow {\log _b}a = c It is the mathematical form of the first part of the consideration. Now coming to the second part bc=a\Rightarrow {b^c} = a It is the mathematical form of the second part. Hope that you got to understand the logarithmic function. ln\ln is a special logarithmic function which has a fixed base equals ee 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 xx using some properties logarithm and
exponential, Now let’s come to the question, we have lnx=7\ln x = - 7 , we can also write this in log\log form as
logex=7{\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 logab{\log _a}b , if we raise logab{\log _a}b in the power of aa then only bb will left
alogab=b\Rightarrow {a^{{{\log }_a}b}} = b

Now we will raise both sides of equation (I) on the power of ee , we will get
elogex=e7 x=e7  \Rightarrow {e^{{{\log }_e}x}} = {e^{ - 7}} \\\ \Rightarrow x = {e^{ - 7}} \\\

We got the required solution for lnx=7\ln x = - 7 which is x=e7x = {e^{ - 7}}
If you want numerical value then calculate this on a calculator you will get a value close to x=9.12×104x = 9.12 \times {10^{ - 4}}

Note: The functions y=ex  &  y=lnxy = {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.