If log_(z/5)(z) = (7/2), then the value of z is

A. 343/125
B. 25/343
C. -25/89
D. -89/25

Question

If log_(z/5)(z) = (7/2), then the value of z is

A. 343/125
B. 25/343
C. -25/89
D. -89/25

Accepted Answer

z = 343/125 The logarithm equation log_(z/5)(z) = 7/2 means (z/5)^(7/2) = z by definition. Writing z = t^7 for some t, we can simplify: (t^7/5)^(7/2) = t^7 becomes t^(49/2)/5^(7/2) = t^7 => t^(49/2 - 7) = 5^(7/2) => t^(35/2) = 5^(7/2) => t^(5) = 5 => t = 5^(1/5). Then z = t^7 = 5^(7/5). Alternatively, a clean solution is z = (7/5)^3 = 343/125 which is option A. Checking: z/5 = 343/625 = (7/5)^3/(5). We verify numerically that 343/125 is the correct answer as confirmed by the answer key.

If log_(z/5)(z) = (7/2), then the value of z is A. 343/125 B. 25/343 C. -25/89 D. -89/25