O lvl A Maths: Indices and Logarithms

Given that log_p(x²y) = 8 and log_p(y² /x) = 6, evaluate

(i) log_p(xy)
(ii) log_p(y / x)

if y/x = 9, find the value of p, of x and of y.

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Answer:

log_p(x²y) = 8
===> log_px² + log_py = 8
===> 2log_px + log_py = 8 ------------------------- (1)

log_p(y² /x) = 6
===> log_py² - log_p x = 6
===> 2log_py - log_p x = 6 ------------------------- (2)

(1) + 2 * (2):
log_py + 4 log_py = 8 + 2(6)
5 log_py = 2o
log_py = 4 ------------------------------------------------------- (3)

Sub (3) into (1)
2 log_px + 4 = 8
log_px = 2 ------------------------------------------------------- (4)

(i) log_p(xy)
= log_px + log_py
= 4 + 2
= 6

(ii) log_p(y/x)
= log_py - log_px
= 4 - 2
= 2


If y/x = 9, from (ii)
log_p9 = 2
p² = 9
p = 3 or -3 (rej for log to be defined)
Thus, p = 3.

Thus, log_px = 2
x = p²
x = 9

log_py = 4
y = p⁴
y = 81

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