2740.11 – Relating Two Logs

If a=log⁡8(225)a = \log_8(225) and b=log⁡2(15)b = \log_2(15), then which of these follows?

  1. a=b2a = \dfrac{b}{2}

  2. a=2b3a = \dfrac{2 b}{3}

  3. a=ba = b

  4. b=a2b = \dfrac{a}{2}

  5. a=3b2a = \dfrac{3b}{2}

Solution

Here we go.

a=log⁡82258a=225=152152=(23)a=23ab=log⁡2152b=15=23a=2(3a2)b=3a2a=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^{\left(\dfrac{3a}{2}\right)} \\b &= \dfrac{3a}{2} \\a &= \dfrac{2b}{3}\end{aligned}

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