Question

How do you find the value of $x$ if ${\log _x}6 = 0.5$ ?

Answer 1

Hint : We will convert the given logarithm into exponential form. After converting in exponential form we need to take the square of both sides. This will give the value of x directly. Or we can use the laws of indices and logs to obtain the value of x.

Complete step-by-step answer :
Given that ${\log _x}6 = 0.5$
We know that $y = {\log _b}x$ can be written as ${b^y} = x$
So we can write the given expression as,
$6 = {x^{0.5}}$
We know that $0.5 = \dfrac{1}{2}$
So we will write above expression as
$6 = {x^{\dfrac{1}{2}}}$
Taking squares on both sides we get,
${\left( 6 \right)^2} = {x^{{{\left( {\dfrac{1}{2}} \right)}^2}}}$
$36 = x$
This is the correct answer.
So, the correct answer is “x=36”.

Note : Note that $y = {\log _b}x$ this is logarithmic form. We can use alternate methods of using the rules of exponential form.
Given that ${\log _x}6 = 0.5$
Now we know that ${\log _b}x$ can be written as $\dfrac{{\log x}}{{\log b}}$ . So let’s write.
$\dfrac{{\log 6}}{{\log x}} = 0.5$
Taking $\log x$ on other side we get,
$\log 6 = 0.5\log x$
We know that $a\log x = \log {x^a}$
$\log 6 = \log {x^{0.5}}$
Cancelling logs on both sides we get,
$6 = {x^{0.5}}$
Now onwards the process is the same as above. That is on squaring we get,
$36 = x$
This is the correct answer.