Given that `10^(0.48) =x` and `10^(0.70)=y` and `x^(z) = y^(2)`, then find the approximate value of z?

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step-by-Step Solution:

1. **Identify the given values:**
We are given:
- \( x = 10^{0.48} \)
- \( y = 10^{0.70} \)
- The equation \( x^z = y^2 \)

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

3. **Apply the power of a power property:**
Using the property \( (a^m)^n = a^{m \cdot n} \), we can simplify both sides:
\[ 10^{0.48z} = 10^{0.70 \cdot 2} \]

4. **Calculate \( 0.70 \cdot 2 \):**
\[ 0.70 \cdot 2 = 1.40 \]
So, the equation becomes:
\[ 10^{0.48z} = 10^{1.40} \]

5. **Set the exponents equal to each other:**
Since the bases are the same, we can set the exponents equal:
\[ 0.48z = 1.40 \]

6. **Solve for \( z \):**
To find \( z \), divide both sides by \( 0.48 \):
\[ z = \frac{1.40}{0.48} \]

7. **Perform the division:**
Calculate \( \frac{1.40}{0.48} \):
\[ z \approx 2.9167 \]

8. **Round to approximate value:**
The approximate value of \( z \) is:
\[ z \approx 2.9 \]

### Final Answer:
The approximate value of \( z \) is \( 2.9 \).

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