How do you solve ${{\log }_{2}}3=x$ ? \[\]

Hint: We recall the definition of logarithm and its different identities. We use the base change identity from ${{\log }_{b}}x=\dfrac{{{\log }_{d}}x}{{{\log }_{d}}b}$ to change the base to base of common logarithm that is 10. We use known values of common logarithm $\log 2=0.3010,\log 3=0.4771$ to find the required value.

Complete step by step answer:
We know that the logarithm is the inverse operation to exponentiation. 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$x$, which means if ${{b}^{y}}=x$ then the logarithm denoted as log and calculated as
\[{{\log }_{b}}x=y\]
Here $x,y,b$ are real numbers subjected to a condition where the argument of logarithm is $x$ which is always positive. The base is also positive and never equal to 1.
We also know that we can change the base of logarithm $b$to the base $d$ using the following identity
\[{{\log }_{b}}x=\dfrac{{{\log }_{d}}x}{{{\log }_{b}}x}\]
We are asked in the question to solve ${{\log }_{2}}3=x$ which means we have to find the value of $x$. We know the common logarithm values of $\log 2,\log 3$ where the base is 10. So we need to convert the base of the given logarithm from 2 to 10. We use the base change formula for $b=2,x=3,d=10$ to have
\[{{\log }_{2}}3=\dfrac{{{\log }_{10}}3}{{{\log }_{10}}2}=\dfrac{\log 3}{\log 2}\]
We put $\log 2=0.3010,\log 3=0.4771$ in the above step to have
\[{{\log }_{2}}3=\dfrac{{{\log }_{10}}3}{{{\log }_{10}}2}=\dfrac{\log 3}{\log 2}=\dfrac{0.4771}{0.3010}\approx 1.585\]
So the solution up to 4 significant digits as $x=1.585$.\[\]

Note:
We note to remember the logarithmic identity involving product as ${{\log }_{b}}\left( mn \right)={{\log }_{b}}m+{{\log }_{b}}n$ and the logarithmic identity involving quotient as ${{\log }_{b}}\left( \dfrac{m}{n} \right)={{\log }_{b}}m-{{\log }_{b}}n$ and when argument and base is equal we have${{\log }_{b}}b=1$. We note that if the base is 2 the logarithm is called binary logarithm and if the base is 10 the logarithm is called common logarithm.