Question

How do you solve \[{{2}^{x}}=3\]?

Answer 1

Hint: This question is from the topic of pre-calculus. In this question, we will use some logarithmic functions to solve. During solving this question, we will first take log to the both sides of the equation. After that, we will use the formula \[\log {{a}^{n}}=n\log a\] and solve the further equation. After solving, we will use the values of \[\log 2\] and \[\log 3\] that are 0.301 and 0.477 respectively in the solving process. After doing some further process, we will find the value of x.

Complete step by step solution:
Let us solve this question.
In this question, we have asked to solve the equation \[{{2}^{x}}=3\]. Or, we can say that we have to solve the given equation and find the value of x.
The given equation is
\[{{2}^{x}}=3\]
Now, taking ‘log’ to the both side of the equation, we can write the above equation as
\[\Rightarrow \log {{2}^{x}}=\log 3\]
Now, using the formula of logarithms \[\log {{a}^{n}}=n\log a\], we can write the above equation as
\[\Rightarrow x\log 2=\log 3\]
Now, dividing \[\log 2\] to the both side of the equation, we can write the above equation as
\[\Rightarrow x=\dfrac{\log 3}{\log 2}\]
We know that the value of \[\log 2\] is \[0.301\] and the value of \[\log 3\] is \[0.477\].
After putting the value of \[\log 2\] and \[\log 3\] in the above equation, we can write
\[\Rightarrow x=\dfrac{0.477}{0.301}\]
After dividing 0.477 by 0.301, we get 1.5847. so, we can write the above equation as
\[\Rightarrow x=1.5847\]
So, now we have solved the equation \[{{2}^{x}}=3\], and found the value of x as 1.5847

Note: We should have a better knowledge in the topic of pre-calculus. We should know about logarithmic functions to solve this type of question easily. We should remember the formula of logarithmic function that is \[\log {{a}^{n}}=n\log a\]. We should remember the following values:
\[\log 2=0.301\]
\[\log 3=0.477\]
Keep in mind that these are approximate values and not the actual value. We can take these values for solving questions. And remember that the base of the above log that is \[\log 2\] and \[\log 3\] is 10.
So, we can say that
\[{{\log }_{10}}2=0.301\]
\[{{\log }_{10}}3=0.477\]