# What is ${\log _a}(\dfrac{1}{a})$?

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Hint: A quantity representing the power to which a fixed number (the base) must be raised to produce a given number. A common and general logarithm is one with base 10, in such cases we don’t have to mention the value of the base. We use certain basic properties of logarithm such as the exponential property which states that:
${\log _b}{m^n} = n{\log _b}m$
This property holds true if and only if $m > 0$.(domain of a logarithmic function)

The given question is: ${\log _a}(\dfrac{1}{a})$
We can write $\dfrac{1}{a} = {a^{ - 1}}$
based on the exponential property ${\log _b}{m^n} = n{\log _b}m$
${\log _a}{a^{ - 1}} = - 1({\log _a}a)$
Which implies: ${\log _a}a = 1$
$( - 1) \times (1) = - 1$
The above synthesis is valid if and only if $a > 0$