Given:
log a = 1/2 log b
log a = 1/5 log c

We need to find the value of a^4b^3c^-2.

Solution:

To find the value of a^4b^3c^-2, we need to simplify the given expressions and substitute them into the equation.

Simplifying the given expressions:

Given log a = 1/2 log b, we can rewrite it as:
log a = log b^(1/2)
Using the logarithmic property, we can rewrite it as:
a = b^(1/2)

Given log a = 1/5 log c, we can rewrite it as:
log a = log c^(1/5)
Using the logarithmic property, we can rewrite it as:
a = c^(1/5)

Substituting the expressions into the equation:

Now, we can substitute the above expressions into the equation a^4b^3c^-2:

a^4b^3c^-2 = (b^(1/2))^4 * b^3 * (c^(1/5))^-2

Simplifying the expression:

Let's simplify each part of the expression step by step.

1. (b^(1/2))^4 = b^(4 * 1/2) = b^2

2. (c^(1/5))^-2 = c^(-2 * 1/5) = c^(-2/5)

Substituting these simplified expressions back into the original equation, we have:

a^4b^3c^-2 = b^2 * b^3 * c^(-2/5)

Using the properties of exponents, we can combine the like terms:

a^4b^3c^-2 = b^(2+3) * c^(-2/5)

Simplifying further, we get:

a^4b^3c^-2 = b^5 * c^(-2/5)

Therefore, the value of a^4b^3c^-2 is b^5 * c^(-2/5).

Final Answer:

So, the value of a^4b^3c^-2 is b^5 * c^(-2/5).