How do you solve the given expression, the expression is \[{3^{2x + 1}} = 5\] ?

Hint: Here for such question in which the power to a number is in linear equation form then either you can differentiate the terms or use log function both the sides, we know that when we use the log function then power will multiply with the log function and we know the value of log for three and five, so here we have to apply log both the side.

Complete step by step solution:
The given question is \[{3^{2x + 1}} = 5\]. Here we have to take log on the both the side of equation, we know the log property which is:
\[\log {a^b} = b\log a\]
Applying this to our expression we get:
\[{3^{2x + 1}} = 5 \]
Taking logarithm on both sides
\[\Rightarrow \log {3^{2x + 1}} = \log 5 \\
\Rightarrow (2x + 1)\log 3 = \log 5 \\
\Rightarrow (2x + 1) \times 0.4771 = 0.6989(since\,\log 3 = 0.4771\,and\,\log 5 = 0.6989) \\
\Rightarrow 0.9542x + 0.4771 = 0.6989 \\
\Rightarrow 0.9542x = 0.6989 - 0.4771 = 0.2218 \\
\therefore x = \dfrac{{0.2281}}{{0.9542}} = 0.2390\]
Hence we got the value of the variable in our expression, which is our final result.

Note: Log function has its own properties for solving the question, and exponential questions can easily be solved by using this function of math. You can also draw graph of log and then find the value of the numbers in log function, or you have to use log table to see the values, mostly log values are very easy to remember and if you remember for two, three, and five then you can easily solve questions.The given question can also be solved by differentiation but that would be lengthy process and answer of the question will match this answer, for differentiation also we have to take the log function first, and then we can apply differentiation.