Problem:
The solution of the equation 7x+7=8x can be expressed in the form x=logb77. What is b?
Answer Choices:
A. 157
B. 87
C. 78
D. 815
E. 715
Solution:
If x=logb77, then bx=77. Thus
(7b)x=7x⋅bx=7x+7=8x
Because x>0, it follows that 7b=8 and so b=78.
\section*{OR}
Taking the logarithm of both sides gives us (x+7)log7=xlog8. Solving, we have xx+7=log7log8,xlog8=xlog7+7log7,x(log8−log7)=7log7, and we have x=log78log77. Using the change of base rule for logarithms, b=78.
The problems on this page are the property of the MAA's American Mathematics Competitions