Simplify: log43 × log24364Choices:- 3/5 '2/5 '3/4 '1/3

The given expression is log4(3) × log24(364).

Step 1: We know that 4 = 2^2 and 24 = 2^3 * 3. So, we can rewrite the expression as log2^2(3) × log2^3*3(364).

Step 2: Using the properties of logarithms, we can bring the powers in front of the logarithms. The expression becomes 2log2(3) × 3log2*3(364).

Step 3: We know that loga(a) = 1. So, log2(2) = 1 and log3(3) = 1. We can simplify the expression to 2log2(3) × 3log2(364).

Step 4: Now, we can use the property of logarithms that loga(bc) = loga(b) + loga(c). So, the expression becomes 2log2(3) + 3*log2(364).

Step 5: We know that 364 = 491 = 4713 = 2^2713. So, we can rewrite the expression as 2log2(3) + 3log2(2^27*13).

Step 6: Using the properties of logarithms, we can bring the powers in front of the logarithms. The expression becomes 2log2(3) + 32log2(713).

Step 7: Now, we can use the property of logarithms that loga(bc) = loga(b) + loga(c). So, the expression becomes 2log2(3) + 6log2(7) + 6log2(13).

Step 8: We can simplify the expression to 2/3 + 6/7 + 6/13.

Step 9: Adding these fractions, we get 2/3 + 6/7 + 6/13 = 2/5.

So, the simplified form of the given expression is 2/5. Therefore, the correct choice is '2/5.