Given that `10^(0.48) = x , 10^(0.70) = y , and x^(z) = y^(2)` , then the value of z is close to

Options

A. `1.45`
B. `1.88`
C. `2.9`
D. `3.7`

Answer

To solve the problem step by step, let's start by rewriting the given equations and then manipulate them to find the value of \( z \).

### Step 1: Rewrite the given equations
We have:
1. \( 10^{0.48} = x \)
2. \( 10^{0.70} = y \)
3. \( x^z = y^2 \)

### Step 2: Substitute the values of \( x \) and \( y \)
Substituting the values of \( x \) and \( y \) into the equation \( x^z = y^2 \):
\[ (10^{0.48})^z = (10^{0.70})^2 \]

### Step 3: Simplify the equation
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \), we can simplify both sides:
\[ 10^{0.48z} = 10^{1.40} \]
Here, \( (10^{0.70})^2 = 10^{0.70 \cdot 2} = 10^{1.40} \).

### Step 4: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 0.48z = 1.40 \]

### Step 5: Solve for \( z \)
Now, we can solve for \( z \):
\[ z = \frac{1.40}{0.48} \]

### Step 6: Calculate the value of \( z \)
Calculating the fraction:
\[ z = \frac{1.40}{0.48} = \frac{140}{48} = \frac{35}{12} \]
Now, we can convert \( \frac{35}{12} \) to a decimal:
\[ \frac{35}{12} \approx 2.9167 \]
So, \( z \) is approximately \( 2.92 \).

### Conclusion
The value of \( z \) is close to \( 2.9 \).

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