Problem Analysis:
We are given that log 5 = 0.6990. We need to find log 125 and log 500 using this information. To solve this problem, we will use the properties of logarithms and the rules of exponents.

Solution:

Step 1: Understanding the relationship between logarithms and exponents
Logarithms are the inverse operation of exponentiation. The logarithm of a number to a given base is the exponent to which the base must be raised to obtain that number. In other words, if b^x = y, then log_b(y) = x.

Step 2: Using the properties of logarithms
The logarithm of a product is equal to the sum of the logarithms of the individual factors. Similarly, the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator.

Step 3: Finding log 125
To find log 125, we need to express 125 as a power of 5 because we know the logarithm of 5. Let's express 125 as 5^x:

5^x = 125

To find x, we can take the logarithm of both sides of the equation:

log(5^x) = log(125)

Using the property of logarithms mentioned in step 2, we can rewrite the equation as:

x * log(5) = log(125)

Substituting the given value of log(5) as 0.6990, we have:

x * 0.6990 = log(125)

Solving for x, we get:

x = log(125) / 0.6990

Therefore, log 125 is equal to log(125) / 0.6990.

Step 4: Finding log 500
To find log 500, we can use the same approach as we did for log 125. Let's express 500 as a power of 5:

5^y = 500

Taking the logarithm of both sides:

log(5^y) = log(500)

Using the property of logarithms mentioned in step 2:

y * log(5) = log(500)

Substituting the given value of log(5), we have:

y * 0.6990 = log(500)

Solving for y, we get:

y = log(500) / 0.6990

Therefore, log 500 is equal to log(500) / 0.6990.

Step 5: Calculating the values of log 125 and log 500
To find the exact values of log 125 and log 500, we need to substitute the respective expressions we found in step 3 and step 4 into a calculator or use logarithm tables.