Question

How do you evaluate ${\log _{\dfrac{1}{3}}}27$?

Answer 1

Hint: In this question we have to evaluate the given expression which is a logarithmic function, to do this first write the number 27 in terms of 3 by prime factorisation, then by using logarithmic properties such as ${\log _{{a^x}}}b = \dfrac{1}{x}{\log _a}b$ and ${\log _a}{x^n} = n{\log _a}x$ we will get an expression and then further simplification of the expression we will get the required result.

Complete step by step solution:
A logarithm is an exponent which indicates to what power a base must be raised to produce a given number.
 $y = {b^x}$exponential form,
$x = {\log _b}y$ logarithmic function, where $x$ is the logarithm of $y$ to the base $b$, and${\log _b}y$ is the power to which we have to raise $b$ to get $y$, we are expressing $x$ in terms of $y$.
Now given expression is ${\log _{\dfrac{1}{3}}}27$,
Convert logarithm i.e., 27 in terms of 3 by prime factorisation, we get,
$ \Rightarrow 27 = 3 \times 3 \times 3 = {3^3}$,
Now rewriting the expression we get,
$ \Rightarrow {\log _{\dfrac{1}{3}}}27 = {\log _{\dfrac{1}{3}}}{\left( 3 \right)^3}$,
This can be written as,
$ \Rightarrow {\log _{\dfrac{1}{3}}}27 = {\log _{{3^{ - 1}}}}{\left( 3 \right)^3}$,
Now, using logarithms property ${\log _{{a^x}}}b = \dfrac{1}{x}{\log _a}b$, so here $a = 3$, $x = - 1$, and $b = 3$, and by substituting the values in the identity we get,
$ \Rightarrow {\log _{\dfrac{1}{3}}}27 = \dfrac{1}{{ - 1}}{\log _3}{\left( 3 \right)^3}$,
Now simplifying we get,
$ \Rightarrow {\log _{\dfrac{1}{3}}}27 = - {\log _3}{\left( 3 \right)^3}$,
Now, using logarithms property ${\log _a}{x^n} = n{\log _a}x$, so here $a = 3$, $x = 3$, and $n = 3$, and by substituting the values in the identity we get,
$ \Rightarrow {\log _{\dfrac{1}{3}}}27 = - 3{\log _3}\left( 3 \right)$,
Now we know that ${\log _a}a = 1$, we get,
$ \Rightarrow {\log _{\dfrac{1}{3}}}27 = - 3\left( 1 \right)$,
Further simplification we get,
$ \Rightarrow {\log _{\dfrac{1}{3}}}27 = - 3$.
So, the value of the given expression is –3.
Final Answer:
$\therefore $The value of the expression i.e., ${\log _{\dfrac{1}{3}}}27$ will be equal to -3.

Note: A logarithm is a mathematical operation that determines how many times a certain number, called the base, is multiplied by itself to reach another number, in these types of questions, we use logarithmic properties and formulas, and some of useful formulas are:
 ${\log _a}xy = {\log _a}x + {\log _a}y$,
${\log _a}{x^n} = n{\log _a}x$,
${\log _a}b = \dfrac{{{{\log }_e}b}}{{{{\log }_e}a}}$,
${\log _{\dfrac{1}{a}}}b = - {\log _a}b$,
${\log _a}a = 1$,
${\log _{{a^x}}}b = \dfrac{1}{x}{\log _a}b$.