Question

If log₁₀ a = m, log₁₀ b = n and K = 10^{5m – 4n}, express K in terms of a and b.

Answer 1

Given: log₁₀ a = m, log₁₀ b = n, K = 10^{5m – 4n}

To find: Express K in terms of a and b.

Calculation:

Step 1: Express 10^5m in terms of a.

Since log₁₀ a = m, we can express a as:

a = 10^m

Therefore:

10^5m = (10^m)⁵ = a⁵

Step 2: Express 4n in terms of b.

Since log₁₀ b = n, we can express b as:

b = 10ⁿ

Thus:

4n = 4 log₁₀ b = log₁₀ b⁴

So:

4n = log₁₀ b⁴

Step 3: Substitute in the equation for K:

Now substitute these expressions into the equation for K:

K = a⁵ – log₁₀b⁴

Thus, K in terms of a and b is:

K = a⁵ – log₁₀b⁴