The formula for converting $a^{5/\log_9a}$ to Base 14 is: $a^{5/\log_9a} = a^{5 \times \frac{\log_{14}a}{\log_{14}9}}$
The significance of using Base 14 in this conversion is that it allows for a more efficient representation of numbers. Unlike the commonly used Base 10, which only has 10 digits, Base 14 has 14 digits, making it possible to represent larger numbers with fewer digits.
To convert a number from Base 10 to Base 14, you can use the repeated division method. Divide the number by 14 and note the remainder. Continue dividing the quotient by 14 until you reach a quotient of 0. The remainders, read from bottom to top, will give you the number in Base 14.
The purpose of the logarithm in this conversion is to determine the number of times a base number (in this case, 9) is multiplied by itself to obtain the original number (a). In other words, it helps us find the exponent that is needed to raise the base to in order to get the original number.
This conversion can be applied to any positive number, as long as the base (a) and the base of the logarithm (9) are both positive and not equal to 1. However, it is important to note that the resulting number may not always be a whole number, depending on the values of a and 9.