Problem: Evaluate 4 log 8/25 .3 log 16/125 .log 5.

Solution:
To solve this problem, we will use the following logarithmic identities:

- log a/b = log a - log b
- log a^n = n log a
- log ab = log a + log b

Now, let's begin solving the problem step by step:

Step 1: Simplify 4 log 8/25
Using the first identity, we get:
4 log 8/25 = 4 (log 8 - log 25)
Using the second identity, we can simplify the logarithms:
4 (log 2^3 - log 5^2) = 4 (3 log 2 - 2 log 5)
= 12 log 2 - 8 log 5

Step 2: Simplify 0.3 log 16/125
Using the first identity, we get:
0.3 log 16/125 = 0.3 (log 16 - log 125)
Using the second identity, we can simplify the logarithms:
0.3 (log 2^4 - log 5^3) = 0.3 (4 log 2 - 3 log 5)
= 1.2 log 2 - 0.9 log 5

Step 3: Simplify log 5
There is no need to simplify this logarithm, so we can leave it as it is.

Step 4: Combine the simplified logarithms
Using the third identity, we can combine the logarithms we obtained in Steps 1, 2 and 3:
12 log 2 - 8 log 5 + 1.2 log 2 - 0.9 log 5 + log 5
= 13.2 log 2 - 8.9 log 5

Step 5: Evaluate the expression
Now we just need to substitute log 2 and log 5 with their respective values:
log 2 ≈ 0.301
log 5 ≈ 0.699

Substituting these values, we get:
13.2 (0.301) - 8.9 (0.699) ≈ 0.702

Therefore, the value of the expression 4 log 8/25 .3 log 16/125 .log 5 is approximately 0.702.