2740.11 – Relating Two Logs

If a=log⁡8225a = \log_8{225} and b=log⁡215b = \log_2{15}, then which of these follows?

  1. a=b2a = \frac{b}{2}
  2. a=2b3a = \frac{2 b}{3}
  3. a=ba = b
  4. b=a2b = \frac{a}{2}
  5. a=3b2a = \frac{3b}{2}
Solution

Here we go.

a=log⁡8225,8a=225=152,152=(23)a=23a,b=log⁡215,2b=15=23a=23a2,b=3a2,a=2b3.\begin{aligned} a &=& \log_8{225}, \\ 8^a &=& 225 = 15^2, \\ 15^2 &=&(2^3)^a = 2^{3a}, \\ b &=& \log_2{15}, \\ 2^b &=& 15 = \sqrt{2^{3a}} = 2^{\frac{3a}{2}}, \\ b &=& \frac{3a}{2}, \\ a &=& \frac{2b}{3}. \end{aligned}

The answer is (b). Stella just couldn't get enough of these problems--loved them.