For what values of the constant A does the equation A-9^-t=0 have one solution?

Question

Solving exponential equations

Answer 1

A-9^-t=0, 9^-t=A. Since 9^-t=1/9^t which is always strictly positive, therefore A>0.

Also -tlog(9)=log(A), t=-log(A)/log(9). If the base of the logs is 9: t=-log₉(A). Logs can only be evaluated for strictly positive arguments. This equation is a linear relationship between variable t and log₉(A). This is a 1-to-1 mapping, implying that there is only one value of t for every value of A>0 and vice versa.