Question

How do you evaluate ${{\log }_{8}}\left( -64 \right)$?

Answer 1

Hint: We are a logarithmic function with base 8 and a negative argument. In order to evaluate this expression, we must have prior knowledge of the logarithmic functions and their properties. Firstly, we will separate the negative sign from the term and then we will make the base and the argument to be represented by the same number so that we can further use logarithmic properties to evaluate the expression.

Complete step-by-step solution:
The logarithmic function has two components namely, a base and an argument.
Given that, ${{\log }_{8}}\left( -64 \right)$. Here, we have the base equal to 8 and the argument equals to -64.
This expression can also be written as ${{\log }_{8}}64\left( -1 \right)$.
$\Rightarrow {{\log }_{8}}\left( -64 \right)={{\log }_{8}}64\left( -1 \right)$
In order to evaluate the given expression, we shall assume and equate it to some constant ‘n’ and then calculate the value of constant-n to find the final solution of given expression.
 $\Rightarrow {{\log }_{8}}\left( -64 \right)=n$
In the formed equation with logarithm function on the left hand side and constant term on right hand side, we shall take the antilogarithm.
$\Rightarrow \left( -64 \right)={{8}^{n}}$
We know that $8\times 8=64$ and not -64.
Therefore, there is no value of ‘n’ which can satisfy the above equation.
Hence, we conclude that ${{\log }_{8}}\left( -64 \right)$ has no solution.

Note: The basic knowledge of logarithmic properties is prominent to solve common problems apart from those like the one given here. From the graph of the logarithm function we must always remember that the logarithm function does not exist and is not defined for negative real numbers and their domain lies only in the interval $\left( 0,\infty \right)$. Also, the y-axis acts like a vertical asymptote for the logarithm function.