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.