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Find the value of \[{\log _{\sqrt 2 }}(\,256\,)\] ?
  A\,)\,8 \\
  B\,)\,4 \\
  C\,)\,15 \\
  D\,)\,16 \\

Last updated date: 14th Apr 2024
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MVSAT 2024
33.6k+ views
Hint: First find the format of the given question as possible. After that find the value of $a$and $b$ from the given equation. Then assign any variable for finding value. Substitute the known values in the equation and take logarithmic and square roots in the equation, the value of the unknown will be known.

Useful Formula:
The given equation is in the form ${\log _a}\,b$, Let us assign the value equal to the given equation in $x$. Thus the given equation is in the form $x\, = \,{\log _a}\,b$. The common formula of squaring the value and applying square root for the value is used.

Complete step by step solution:
Given that: \[{\log _{\sqrt 2 }}(\,256\,)\,\,\,\, \to \,(\,1\,)\]
The given equation is in the form ${\log _a}\,b$.

We want to find the equivalent value for the given equation.
Let us assume that the equivalent value for the given equation is $x$.
Thus, the equation should be as follows:
$x\, = \,{\log _a}\,b\,\,\,\, \to \,(\,2\,)$
Now compare the given equation and the assumed equation:
$\,{\log _a}\,b\, = \,{\log _{\sqrt 2 }}\,(256)$
With the help of above equation, we need to find the value for $a$ and $b$ as follows:
$\,{\log _a}\,b\, = \,{\log _{\sqrt 2 }}\,(256)$
The value of $a$ is $\sqrt 2 $ and the value of $b$ is $256$

Now, simplify the equation $(\,2\,)$ as possible to get the value of $x$
$x\, = \,{\log _a}\,b\,\,\,$
Now, apply the logarithmic function to both Left hand side and Right-hand side
${b^x}\, = \,a$
The above equation is obtained by cancelling the $\log $ value on the right side and the $x$ will become the power value for $b$.

Now, apply the value of $a$ and $b$ in the equation ${b^x}\, = \,a$, as follows
The value of $a$ is $\sqrt 2 $ and the value of $b$ is $256$
Thus, the equation becomes as follows:
${b^x}\, = \,a$
${(\,256\,)^x}\, = \,\sqrt 2 $
Multiply and divide with power of $x$ in both sides to simplify the equation as follows:
$256\, = \,{(\sqrt 2 )^x}$

Now simplify the value of $256$ in the term of $2$ with the corresponding power value
Thus the $256$ becomes ${2^8}$.
Now substitute the value ${2^8}$ instead of $256$ in the equation.
${2^8}\, = \,{(\sqrt 2 )^x}$
Now multiply with the square root value for left hand side in the above equation:
${(\sqrt 2 )^{16}}\, = \,{(\sqrt 2 )^x}$
Cancel the $\sqrt 2 $value from both sides, to get the value of $x$.
$x = \,16$
We find the equivalent value for the given equation as $x\, = \,16$, where $x$ is the assumed equivalent value.

Thus, the option (D) is the correct answer for the given equation.

Note: The value ${b^x}\, = a$ is the equivalent value of $b\, = \,{a^x}$.This is obtained by multiplying and dividing the value of power of $x$. While applying square root to the value, the power value for the corresponding value will become twice.