#Logarithms: calculating from 2 others, and then proof.

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feral tangle
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The above is the question, and it's non-calculator. At the moment I'm just considering the first bit of estimating them, but I've kinda ran into a wall. There isn't any mark scheme (this is a past STEP paper), so I'd be grateful if anyone could just sort of inform me how to go about this.

My first thought was that (excuse the random nature of my scribblings please haha, also using log base 10):
I should calculate log(6) , which is equal to log(2) + log(5) due to the laws. I should then find log(4) by adding log(2)+log(2). And then I find the halfway point between it. Is that right?
I'm not sure how to go about the inequality part at all however, especially without a calculator. Grateful for any assistance , thanks 🙂

stoic spokeBOT