Question

How do you find the exact value of ${\log _2}\sqrt[4]{8}$?

Answer 1

Hint: In this question we have to find the value of logarithmic function. Logarithm is the inverse function of exponential function. That means the logarithm of a given number $x$ is the exponent to which another fixed number, the base $b$, must be raised, to produce that number $x$. For example, the logarithmic function $y = {\log _b}x$ is defined to be equivalent to the exponential function $x = {b^y}$.

Complete step by step answer:
To find the exact value of the given function ${\log _2}\sqrt[4]{8}$ we will use two laws, law of logarithms and law of exponents.Law of logarithm states that the logarithm of a given number $x$ is the exponent to which another fixed number, the base $b$, must be raised, to produce that number $x$. If $y = {\log _b}x$ then $x = {b^y}$.

According to the law of exponents, in a fractional exponent, the numerator is the power and the denominator is the root. For example, ${x^{\dfrac{a}{b}}} = \sqrt[b]{{{x^a}}}$.Now, we use both the laws to find the exact value of the given logarithm function.
We have, ${\log _2}\sqrt[4]{8}$
Using the law of logarithm
${\log _b}x = y\, \Rightarrow x = {b^y}$
We can reduce the given function by using the above formula as
$ \Rightarrow {\log _2}\sqrt[4]{8}\, = y$
$ \Rightarrow \sqrt[4]{8} = {2^y}$

Now, using the law of exponents. We get,
${x^{\dfrac{a}{b}}} = \sqrt[b]{{{x^a}}}$
$ \Rightarrow \sqrt[4]{8} = \sqrt[4]{{{2^3}}}$
We can reduce this equation by the above formula as
$ \Rightarrow \sqrt[4]{{{2^3}}} = {2^{\dfrac{3}{4}}} = {2^y}$
Since the bases on both sides are $2$ we can equate the exponents.
$ \Rightarrow y = \dfrac{3}{4}$
$ \therefore {\log _2}\sqrt[4]{8} = \dfrac{3}{4}$

Hence, we get the exact value of the given function ${\log _2}\sqrt[4]{8}$ equal to $\dfrac{3}{4}$.

Note: A fractional exponent is represented as ${x^{\dfrac{a}{b}}}$ where $x$ is a base and $\dfrac{a}{b}$ is an exponent. This expression is equivalent to the ${b^{^{th}}}$ root of $x$ raised to the ${a^{th}}$ power, or $\sqrt[b]{{{x^a}}}$. In fractional exponent, the exponent is written before the radical symbol.