Question

How do you solve \[\ln 2 + \ln x = 5\]?

Answer 1

Hint: In the given question, we have been given an expression. It is an expression of a natural logarithm. There are two terms of the natural logarithm. They are separated by the plus sign. It is equal to a constant. We have to simplify the expression. We can easily do that if we know the properties of the logarithm.

Formula Used:
We are going to use the sum formula of natural logarithm:
\[\ln a + \ln b = \ln \left( {a \times b} \right)\]

Complete step by step answer:
The given expression is
\[\ln 2 + \ln x = 5\]
To solve this question, we are going to use the sum formula of natural logarithm, which is,
\[\ln a + \ln b = \ln \left( {a \times b} \right)\]
Substituting \[a = 2\] and \[b = x\], we have,
\[\ln 2 + \ln x = \ln 2x\]
Now, \[\ln 2x = 5\]
Raising both sides to the power of \[e\], we have,
\[{e^{\ln 2x}} = {e^5}\]
Now, we know, \[{e^{\ln nx}} = nx\]
So, \[2x = {e^5}\]

Hence, \[x = {e^5}/2\]

Additional Information:
The \[\log \] function has other basic properties too:
\[{\log _x}{x^n} = n\]
\[{\log _b}a = n \Rightarrow {b^n} = a\]
\[{\log _a}b = \dfrac{1}{{{{\log }_b}a}}\]

Note:
In this question, to solve for the answer, we needed to know the properties of the logarithmic function. We needed to know how \[\ln \] and \[\log \] are related. If we know such basic things of any topic, we can easily solve for the answer.