If x^a = y^b = k and x^c = y^d = t, then which of the following is true?
a) ac = bd
b) ad = bc
c)a/d = c/b
d) a^c = b^d
e) a+c = b+d
Ya I need help with this too
To determine which of the given statements is true, let's break down the information provided.
We are given two equations:
1) x^a = y^b = k
2) x^c = y^d = t
To begin, we can solve both equations for x and y individually.
From equation 1:
x^a = k
Taking the logarithm of both sides (any base will work, let's use base 10):
log(x^a) = log(k)
a * log(x) = log(k)
log(x) = log(k) / a
x = 10^(log(k) / a)
From equation 2:
x^c = t
Taking the logarithm of both sides (again using base 10):
log(x^c) = log(t)
c * log(x) = log(t)
log(x) = log(t) / c
x = 10^(log(t) / c)
Similarly, we can solve for y using the same approach:
y = 10^(log(k) / b) from equation 1
y = 10^(log(t) / d) from equation 2
Now, let's substitute the expressions for x and y back into the given statements to simplify and determine which statement is true.
a) ac = bd
Substituting x and y:
(log(k) / a) * (log(t) / c) = (log(k) / b) * (log(t) / d)
(log(k)^2 / ac) = (log(k)^2 / bd)
ac = bd
b) ad = bc
Substituting x and y:
(log(k) / a) * (log(t) / d) = (log(k) / b) * (log(t) / c)
(log(k) / a) * (log(t) / d) = (log(k) / b) * (log(t) / c)
ad = bc
c) a/d = c/b
Substituting x and y:
a/d = c/b
d) a^c = b^d
Substituting x and y:
(10^(log(k) / a))^c = (10^(log(t) / b))^d
(10^(log(k) * c / a)) = (10^(log(t) * d / b))
k^(c / a) = t^(d / b)
Taking the logarithm of both sides:
(c / a) * log(k) = (d / b) * log(t)
c/a = d/b
e) a+c = b+d
a + c = b + d
Based on the calculations, the statement that holds true is:
e) a+c = b+d