If base is not mentioned, then always remember to take it as 10.

Hence, in the given expression, assume base as 10

We are given, $[log_{10} 2 + log_{10} (4x + 1)$ = $log_{10} (x + 2) + 1]$

$[log_{10} 2 + log_{10} (4x + 1)$ = $log_{10} (x + 2) + 1]$

$[log_{10} 2 + log_{10} (4x + 1)$ = $log_{10} (x + 2) + log_{10}10]$

Now, Use the product rule: $log_{a}(xy)$ = $log_{a}x + log_{a}y$

$[log_{10} 2 (4x + 1)]$ = $[log_{10} 10(x + 2)]$

(4x + 1) = (5x + 10)

4x + 1 = 5x + 10

x = - 9