If `(2.4)^(x) =(0.24)^(y) =10^(z)`, then show that `1/x - 1/z =1/y`

Answer

To solve the equation \( (2.4)^x = (0.24)^y = 10^z \) and show that \( \frac{1}{x} - \frac{1}{z} = \frac{1}{y} \), we can follow these steps:

### Step 1: Set the common value
Let \( k = (2.4)^x = (0.24)^y = 10^z \).

### Step 2: Express each base in terms of \( k \)
From the definition of \( k \):
1. \( 2.4 = k^{\frac{1}{x}} \)
2. \( 0.24 = k^{\frac{1}{y}} \)
3. \( 10 = k^{\frac{1}{z}} \)

### Step 3: Divide the first two equations
Now, divide the first equation by the second:
\[ \frac{2.4}{0.24} = \frac{k^{\frac{1}{x}}}{k^{\frac{1}{y}}} \]
This simplifies to:
\[ \frac{2.4}{0.24} = k^{\frac{1}{x} - \frac{1}{y}} \]

### Step 4: Calculate \( \frac{2.4}{0.24} \)
Calculating the left side:
\[ \frac{2.4}{0.24} = 10 \]
Thus, we have:
\[ 10 = k^{\frac{1}{x} - \frac{1}{y}} \]

### Step 5: Use the third equation
From the third equation, we know:
\[ 10 = k^{\frac{1}{z}} \]

### Step 6: Set the exponents equal
Since both expressions equal 10, we can set the exponents equal to each other:
\[ \frac{1}{x} - \frac{1}{y} = \frac{1}{z} \]

### Step 7: Rearranging the equation
Rearranging gives us:
\[ \frac{1}{x} - \frac{1}{z} = \frac{1}{y} \]

This completes the proof that \( \frac{1}{x} - \frac{1}{z} = \frac{1}{y} \).

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