Given that 10^0.48 = x and 10^0.70 = y, and x raised to the power z equals y squared (x^z = y^2), the value of z is closest to which of the following?

Introduction:
This question tests exponent rules and using logarithms (or exponent comparison) to solve for an unknown power. Because x and y are both written as powers of 10, we can rewrite x^z and y^2 as powers of 10 as well. When two powers of 10 are equal, their exponents must be equal. That converts the problem into a simple linear equation in z. The final step is choosing the closest option, because the exact value may not match an option exactly.

Given Data / Assumptions:

• x = 10^0.48
• y = 10^0.70
• Relation: x^z = y^2
• Rule: (a^m)^n = a^(m*n)
• If 10^A = 10^B, then A = B

Concept / Approach:
Rewrite both sides using base 10. Since x = 10^0.48, x^z = (10^0.48)^z = 10^(0.48z). And y^2 = (10^0.70)^2 = 10^(1.40). Equate exponents: 0.48z = 1.40, then solve for z and compare to the choices.

Step-by-Step Solution:
x^z = (10^0.48)^z = 10^(0.48z) y^2 = (10^0.70)^2 = 10^(0.70*2) = 10^1.40 So 10^(0.48z) = 10^1.40 Equate exponents: 0.48z = 1.40 z = 1.40 / 0.48 = 140/48 = 35/12 ≈ 2.9167 Closest option to 2.9167 is 2.9

Verification / Alternative check:
If z ≈ 2.9167, then 0.48z ≈ 0.48*2.9167 ≈ 1.40, so x^z ≈ 10^1.40 which matches y^2 exactly in exponent form.

Why Other Options Are Wrong:
2.7 gives exponent 0.48*2.7 = 1.296, too low. 3.6 and 3.7 give exponents above 1.7, too high. 3.1 gives 1.488, still too high. Only 2.9 is closest to 2.9167.

Common Pitfalls:
Multiplying exponents incorrectly, treating 10^0.70 squared as 10^0.49, or forgetting that equal bases imply equal exponents.

Final Answer:
The value of z is closest to 2.9.