If log_P(Q) = x^2 and log_Q(P)=1/(2x-1) find x.
log_p(q) = 1/log_q(p), so
x^2 = 2x-1
x = 1
did you mean:
logPQ = x^2 and logQ P = 1/(2x-1) find x. ?????
I will assume you do ...
A lesser know property of logs is that
if logab = x
then logb/sub>a = 1/x
that is:
logab = 1/logb/sub>a
e.g. check on your calculator that log5/sub>3 = 1/log3/sub>5
from yours ....
x^2 = 1/[1/(2x - 1) ] = 2x - 1
x^2 - 2x + 1 = 0
(x-1)^2 = 0
x-1 = 0
x = 1
argghhhh, that's what you get by trying to be fancy with code.
good to know it works, just forgot to close the one loop
I always copy/paste the ? pairs, then fill in the contents.
I've been stung before, too!
To find the value of x, we will use the fact that logarithmic functions are inverses of exponential functions. Let's begin by rewriting the given equations:
1. log_P(Q) = x^2
2. log_Q(P) = 1/(2x-1)
Now, let's apply the inverse relationship to equation 1. Remember that the base of the logarithm and the base of the exponential function are the same:
3. P^(x^2) = Q
Similarly, we can apply the inverse relationship to equation 2:
4. Q^(1/(2x-1)) = P
Since equations 3 and 4 both express P in terms of Q, we can equate them:
P^(x^2) = Q^(1/(2x-1))
Now, we need to eliminate the bases P and Q to get rid of variables and simplify the equation. To do this, we can take the logarithm of both sides using any base. Let's use the natural logarithm (ln):
ln(P^(x^2)) = ln(Q^(1/(2x-1)))
Using the property of logarithms that allows us to bring the exponent down as a multiplier:
x^2 * ln(P) = (1/(2x-1)) * ln(Q)
Now, we can substitute the given values into the equation. However, since the values of P and Q are not mentioned in the question, we cannot determine the exact value of x without further information. But we can proceed using general variables for P and Q:
x^2 * ln(P) = (1/(2x-1)) * ln(Q)
Therefore, to find the value of x, we need the specific values of P and Q.