Final answer:
To find logₑ y in terms of a and b, we can use the properties of logarithms. We start by rewriting the given equations using the properties of logarithms. Then, we simplify the equations by combining the logarithms and coefficients. Finally, we isolate logₑ y to find an expression for it in terms of a and b. logₑ y = (b - a - 2logₑ x) / 2
Step-by-step explanation:
To find logₑ y in terms of a and b, we can use the properties of logarithms. We are given that logₑ x⁴y = a and logₑ x²y² = b. Using the property that the logarithm of a product is the sum of the logarithms, we can rewrite the first equation as logₑ x⁴ + logₑ y = a. Similarly, we can rewrite the second equation as logₑ x² + logₑ y² = b.
Using the property that the logarithm of an exponent is the product of the exponent and the logarithm, we can simplify the equations to 4logₑ x + logₑ y = a and 2logₑ x + 2logₑ y = b. Subtracting the first equation from the second equation, we get 2logₑ y = b - a - 2logₑ x. Dividing both sides by 2, we find logₑ y = (b - a - 2logₑ x) / 2. This gives us an expression for logₑ y in terms of a and b.