Question

How do you solve $8\ln x = 1$?

Answer 1

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;
$\Rightarrow x = 1.133$