Example  SEQ.3.  Suppose \$3*2^{x-1}+1= 13\$. Find \$x\$.
Solution: Since \$b=2, A= 3 \$, the equation can be solved by solving  \$3*2^{x-1}= 12\$ or \$2^{x-1}= 4\$. Thus \$x -1 =2\$ and \$x = 3\$.

Comment: We can consider the expression on the left hand side of the equation as a composite function of \$x\$ giving \$  f(x) = 3*2^{x-1}+1\$ . Now the problem can be restated: to find a \$x\$ where \$f(x) = 13\$. This problem and its solution can be visualized with the mapping diagram for the function \$f\$ as the composite of four functions: \$R(x) =x-1 ; exp_2 (x) = 2^x; m(x) = 3*x;\$ and \$s(x)=x+1\$.

Comments: You can move the (red) point labelled x on the left axis of the mapping diagram to a position where the arrow head points to \$f(x) = 13\$.
Check the box and the diagram will show the solution on the composite mapping diagram by reversing the arrows and that when \$s(x) =x+1=13, x=12;\$ when \$m(x) = 3x = 12, x = 4;\$ when \$exp_2(x)= 2^x=4, x =2;\$ and when \$R(x)= x-1 =2; x =3.\$

You can use the sliders to investigate other examples by changing the base, \$A,h\$, and \$k\$, as well as the point on the final target axis where \$C=13\$.