TITLE: Apparently cannot be solved using logarithms
QUESTION [9 upvotes]: This equation clearly cannot be solved using logarithms. 
$$3 + x = 2 (1.01^x)$$
Now it can be solved using a graphing calculator or a computer and the answer is $x = -1.0202$ and $x=568.2993$.
But is there any way to solve it algebraically/algorithmically?

REPLY [15 votes]: I have solved a question similar to this before. In general, you can have a solution of the equation 

$$ a^x=bx+c $$ 

in terms of the Lambert W-function

$$ -\frac{1}{\ln(a)}W_k \left( -\frac{1}{b}\ln(a) {{\rm e}^{-{\frac {c\ln(a)  }{b}}}} \right)-{\frac {c}{b}}
 \,.$$

Substituting $ a=1.01 \,,b=\frac{1}{2}\,,c=\frac{3}{2}$ and considering the values $k=0$ and $k=-1$, we get the zeroes $$x_1= -1.020199952\,, x_2=568.2993002 \,. $$