If `loga=1/2 logb=1/5logc` then `a^4b^3c^(-2)=`

Options

A. a = 24
B. b = 81
C. c= 64
D. c = 256

Explanation

To solve the problem, we need to find the value of \( a^4 b^3 c^{-2} \) given that \( \log a = \frac{1}{2} \log b = \frac{1}{5} \log c \).

### Step 1: Set up the equations
From the given information, we can express the logarithms in terms of a constant \( k \):
\[ \log a = k, \quad \log b = 2k, \quad \log c = 5k \]

### Step 2: Express \( a \), \( b \), and \( c \) in terms of \( k \)
Using the properties of logarithms, we can express \( a \), \( b \), and \( c \) as follows:
\[ a = e^k, \quad b = e^{2k}, \quad c = e^{5k} \]

### Step 3: Substitute into the expression \( a^4 b^3 c^{-2} \)
Now we substitute these values into the expression \( a^4 b^3 c^{-2} \):
\[ a^4 = (e^k)^4 = e^{4k} \]
\[ b^3 = (e^{2k})^3 = e^{6k} \]
\[ c^{-2} = (e^{5k})^{-2} = e^{-10k} \]

### Step 4: Combine the expressions
Now we can combine these results:
\[ a^4 b^3 c^{-2} = e^{4k} \cdot e^{6k} \cdot e^{-10k} \]

### Step 5: Simplify the expression
Using the property of exponents that states \( e^x \cdot e^y = e^{x+y} \):
\[ a^4 b^3 c^{-2} = e^{4k + 6k - 10k} = e^{0} = 1 \]

### Final Answer:
Thus, the value of \( a^4 b^3 c^{-2} \) is \( \boxed{1} \).

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