Let's solve the problem step-by-step.
1. Given Equations:
- [tex]\(10^{0.48} = x\)[/tex]
- [tex]\(10^{0.70} = y\)[/tex]
- [tex]\(x^z = y^2\)[/tex]
2. Convert to Logarithmic Form:
- From [tex]\(10^{0.48} = x\)[/tex], taking the logarithm base 10 on both sides we get:
[tex]\[
\log_{10}(x) = 0.48
\][/tex]
- Similarly, from [tex]\(10^{0.70} = y\)[/tex], taking the logarithm base 10 on both sides we get:
[tex]\[
\log_{10}(y) = 0.70
\][/tex]
3. Express [tex]\(x^z = y^2\)[/tex] in Terms of Logarithms:
- From the given formula [tex]\(x^z = y^2\)[/tex], we take the logarithm base 10 on both sides:
[tex]\[
\log_{10}(x^z) = \log_{10}(y^2)
\][/tex]
- Using the power rule of logarithms ([tex]\(\log_{10}(a^b) = b \log_{10}(a)\)[/tex]), this becomes:
[tex]\[
z \log_{10}(x) = 2 \log_{10}(y)
\][/tex]
4. Solve for [tex]\(z\)[/tex]:
- Substituting the earlier values [tex]\(\log_{10}(x) = 0.48\)[/tex] and [tex]\(\log_{10}(y) = 0.70\)[/tex] into the equation:
[tex]\[
z \cdot 0.48 = 2 \cdot 0.70
\][/tex]
- Simplify:
[tex]\[
z \cdot 0.48 = 1.40
\][/tex]
- Solve for [tex]\(z\)[/tex]:
[tex]\[
z = \frac{1.40}{0.48}
\][/tex]
- This simplifies to:
[tex]\[
z \approx 2.9166666666666665
\][/tex]
5. Determine the Closest Value:
- Comparing [tex]\(2.9166666666666665\)[/tex] to the given choices:
[tex]\[
\text{(a) 1.45}, \quad \text{(b) 1.88}, \quad \text{(c) 2.9}, \quad \text{(d) 3.7}
\][/tex]
- The closest value to [tex]\(2.9166666666666665\)[/tex] is [tex]\(2.9\)[/tex].
Therefore, the value of [tex]\(z\)[/tex] is closest to [tex]\(2.9\)[/tex]. The correct answer is:
[tex]\[
\boxed{2.9}
\][/tex]
