If `a^(2) = log x, b^(3) = log y and (a^(2))/(2) - (b^(3))/(3) = log c`, find c in terms of x and y.

Answer

To solve the problem step by step, we will start with the given equations and manipulate them to find \( c \) in terms of \( x \) and \( y \).

### Step 1: Write down the given equations
We are given:
1. \( a^2 = \log x \)
2. \( b^3 = \log y \)
3. \( \frac{a^2}{2} - \frac{b^3}{3} = \log c \)

### Step 2: Substitute the values of \( a^2 \) and \( b^3 \)
Substituting the values of \( a^2 \) and \( b^3 \) into the third equation:
\[ \frac{\log x}{2} - \frac{\log y}{3} = \log c \]

### Step 3: Find a common denominator
To combine the fractions on the left side, we need a common denominator. The least common multiple of 2 and 3 is 6. Thus, we can rewrite the equation as:
\[ \frac{3 \log x}{6} - \frac{2 \log y}{6} = \log c \]

### Step 4: Combine the fractions
Now we can combine the fractions:
\[ \frac{3 \log x - 2 \log y}{6} = \log c \]

### Step 5: Multiply both sides by 6
To eliminate the fraction, multiply both sides by 6:
\[ 3 \log x - 2 \log y = 6 \log c \]

### Step 6: Rewrite the equation using logarithmic properties
Using the property of logarithms that states \( a \log b = \log(b^a) \), we can rewrite the left side:
\[ \log(x^3) - \log(y^2) = \log(c^6) \]

### Step 7: Use the property of logarithms to combine
Using the property \( \log a - \log b = \log\left(\frac{a}{b}\right) \):
\[ \log\left(\frac{x^3}{y^2}\right) = \log(c^6) \]

### Step 8: Remove the logarithms
Since the logarithms are equal, we can set the arguments equal to each other:
\[ \frac{x^3}{y^2} = c^6 \]

### Step 9: Solve for \( c \)
To find \( c \), take the sixth root of both sides:
\[ c = \left(\frac{x^3}{y^2}\right)^{\frac{1}{6}} \]

### Final Result
Thus, we have:
\[ c = \frac{x^{\frac{1}{2}}}{y^{\frac{1}{3}}} \]