Exponential And Logarithmic Functions

To solve the equation , we will use a substitution method. Let's assume . This substitution transforms the equation into a quadratic form:

Now, this is a standard quadratic equation of the form , where , , and . We can use the quadratic formula to solve for :

Substituting the values of , , and :

This gives us two solutions for :

Reverting the substitution , we have two equations:

Solving for in each equation:

To find the sum of all solutions, we calculate:

• Using the property , we have:

Finally, since , we can express this in base 6:

• Since , we have

Thus, the sum of all solutions is , confirming that the correct answer is: