Given that log₂(3) = a, log₃(5) = b, log₇(2) = c, express the logarithm of the number 63 to the base 140 in terms of a, b, and c.

To express the logarithm of 63 to the base 140 in terms of a, b, and c, we first need to break down the numbers involved using their prime factorization.

Step 1: Factorization of 63 and 140

The number 63 can be factored as:

• 63 = 3^2 × 7

The number 140 can be factored as:

• 140 = 2 × 5 × 7

Step 2: Applying the Change of Base Formula

We want to find log₁₄₀(63). Using the change of base formula, we have:

log₁₄₀(63) = log(63) / log(140)

Step 3: Expressing Logarithms in Terms of a, b, and c

Now, we can express log(63) and log(140) using the known values:

• log(63) = log(3^2) + log(7) = 2 log(3) + log(7)
• log(140) = log(2) + log(5) + log(7)

Step 4: Substituting Known Values

Using the given values:

• log(3) = log₂(3) / log₂(2) = a
• log(5) = log₃(5) / log₃(3) = b
• log(2) = log₇(2) / log₇(7) = c

Final Expression

Now we can substitute these values into our earlier expressions:

• log(63) = 2a + c
• log(140) = c + b + c = c + b + c = 2c + b

Thus, we have:

log₁₄₀(63) = (2a + c) / (2c + b)

This gives us the logarithm of 63 to the base 140 expressed in terms of a, b, and c.