Logarithms

Question

Given that a>b>1 and that 2 log_ab + 4 log_ba=9, find b in terms of a.

 

Ans: b=√a

Answer 1

Use the fact that log_ba = 1/log_ab. Let x = log_ab = 1/log_ba, so your equation becomes

2 x + 4/x = 9.

Multiply by x:

2x² -9x + 4 = 0.

Solve this quadratic equation, get

x = 1/2, 4.

Since a>b, x = log_ab <1, so only x=1/2 is a valid solution.

Now solve

log_ab = 1/2 ⇒ b = a^1/2 = √a.