Step-by-step explanation:
To prove that z = xy / 2x + y when 3^x * 5^y = (75)^z, we can use logarithms to simplify the equation.
Taking the logarithm of both sides of the equation, we get:
log (3^x * 5^y) = log ((75)^z)
Using the properties of logarithms, we can simplify this further:
log (3^x * 5^y) = z * log (75)
Now, let's expand the logarithm on the left side using the properties of logarithms:
log (3^x * 5^y) = log (3^x) + log (5^y)
Using the logarithmic identities, we have:
x * log (3) + y * log (5) = z * log (75)
Now, we can simplify the equation by substituting the values of logarithms:
x * log (3) + y * log (5) = z * (log (3) + log (5))
Expanding further:
x * log (3) + y * log (5) = z * log (3) + z * log (5)
Rearranging the terms:
x * log (3) - z * log (3) = z * log (5) - y * log (5)
Factoring out the common logarithm:
log (3) * (x - z) = log (5) * (z - y)
Now, we can divide both sides of the equation by log (3) * (z - y):
(x - z) / (z - y) = log (5) / log (3)
Using the change of base formula, we can rewrite the right side of the equation:
(x - z) / (z - y) = log base (3) (5)
Since the left side of the equation is in terms of x, y, and z, and the right side of the equation is a constant, we can equate them:
Cross-multiplying:
(x - z) * log base (3) (5) = (z - y)
Expanding:
x * log base (3) (5) - z * log base (3) (5) = z - y
Rearranging terms:
z - y = x * log base (3) (5) - z * log base (3) (5)
Adding y to both sides:
z = x * log base (3) (5) - z * log base (3) (5) + y
z + z * log base (3) (5) = x * log base (3) (5) + y
Factoring out z:
z * (1 + log base (3) (5)) = x * log base (3) (5) + y
Dividing both sides by (1 + log base (3) (5)):
z = (x * log base (3) (5) + y) / (1 + log base (3) (5))
Simplifying further:
z = (xy / x + y) * log base (3) (5) + y / (1 + log base (3) (5))
Therefore, we have shown that z = xy / 2x + y when 3^x * 5^y = (75)^z.
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Verified answer
Step-by-step explanation:
To prove that z = xy / 2x + y when 3^x * 5^y = (75)^z, we can use logarithms to simplify the equation.
Taking the logarithm of both sides of the equation, we get:
log (3^x * 5^y) = log ((75)^z)
Using the properties of logarithms, we can simplify this further:
log (3^x * 5^y) = z * log (75)
Now, let's expand the logarithm on the left side using the properties of logarithms:
log (3^x * 5^y) = log (3^x) + log (5^y)
Using the logarithmic identities, we have:
x * log (3) + y * log (5) = z * log (75)
Now, we can simplify the equation by substituting the values of logarithms:
x * log (3) + y * log (5) = z * (log (3) + log (5))
Expanding further:
x * log (3) + y * log (5) = z * log (3) + z * log (5)
Rearranging the terms:
x * log (3) - z * log (3) = z * log (5) - y * log (5)
Factoring out the common logarithm:
log (3) * (x - z) = log (5) * (z - y)
Now, we can divide both sides of the equation by log (3) * (z - y):
(x - z) / (z - y) = log (5) / log (3)
Using the change of base formula, we can rewrite the right side of the equation:
(x - z) / (z - y) = log base (3) (5)
Since the left side of the equation is in terms of x, y, and z, and the right side of the equation is a constant, we can equate them:
(x - z) / (z - y) = log base (3) (5)
Cross-multiplying:
(x - z) * log base (3) (5) = (z - y)
Expanding:
x * log base (3) (5) - z * log base (3) (5) = z - y
Rearranging terms:
z - y = x * log base (3) (5) - z * log base (3) (5)
Adding y to both sides:
z = x * log base (3) (5) - z * log base (3) (5) + y
Rearranging terms:
z + z * log base (3) (5) = x * log base (3) (5) + y
Factoring out z:
z * (1 + log base (3) (5)) = x * log base (3) (5) + y
Dividing both sides by (1 + log base (3) (5)):
z = (x * log base (3) (5) + y) / (1 + log base (3) (5))
Simplifying further:
z = (xy / x + y) * log base (3) (5) + y / (1 + log base (3) (5))
Therefore, we have shown that z = xy / 2x + y when 3^x * 5^y = (75)^z.