Question

How do you solve \[8\ln x = 1\]?

Answer 1

Hint: Logarithm is the inverse function of exponentiation. It means that the logarithm of a given number \[y\] is the exponent to which another number, the base \[b\], must be raised to produce that number \[y\]. To solve this question, we will shift \[8\] to the RHS. Then we will use the property of logarithm from its definition to find the value of variable \[x\].

Complete step-by-step solution:
Given equation:
\[8\ln x = 1\]
On shifting \[8\] to the RHS we get;
\[ \Rightarrow \ln x = \dfrac{1}{8}\]
Now we have to keep in mind that when \[\log \] is written as \[\ln \], it means that the base of the logarithm is \[e\] and it is called natural logarithm. So, we have;
\[ \Rightarrow {\ln _e}x = \dfrac{1}{8}\]
Now from the definition of logarithm we get;
\[ \Rightarrow x = {e^{\dfrac{1}{8}}}\]
On solution we get;
\[ \Rightarrow x = 1.133\]
Additional Information:
The defining relation between exponentiation and logarithm is:
If, \[{\log _b}x = y\], then, \[x = {b^y}\].
Here, \[x\] is the argument and \[b\] is called the base. We should note that argument and base should be greater than zero and base should not be equal to one. Mathematically,
\[x > 0,b > 0{\text{ and }}b \ne 1\].
When the base of the logarithm is \[10\], it is called decimal or common logarithm and when base is \[e\], it is called natural logarithm. One important formula of logarithm is that:
\[{\log _b}x + {\log _b}y = {\log _b}\left( {xy} \right)\].
Other important property of logarithm is:
\[{\log _b}{x^n} = n{\log _b}x\]

Note: We can also solve this question by using the property that \[{\log _b}{x^n} = n{\log _b}x\].
We have the equation as;
\[8\ln x = 1\]
From the property mentioned above we can write it as;
\[ \Rightarrow \ln {x^8} = 1\]
Now evaluating it using the definition of logarithm we get;
\[ \Rightarrow {x^8} = {e^1}\]
Sifting \[8\] to the exponential of RHS we get;
\[ \Rightarrow x = {e^{\dfrac{1}{8}}}\]
On solving it we get;
\[ \Rightarrow x = 1.133\]