Two-stepping into the log plot

I knew T175 would present me with some mathsy stuff, & was prepared to face it bravely, but I did almost turn & run when it wanted to introduce me to logarithmic scales. I mean, it's one thing meeting old mathematical acquaintances & discovering they're not quite the bogeymen you always thought, but quite another to be whisked unsuspecting into the presence of Strangers. Logarithms for me were (are!) arcane sets of numbers that inhabit the pages of a brown-paper-covered booklet known as Log Tables, whose name is a cruel deception: no resemblance at all to picnics. But logarithmic scales? Nope, never heard of 'em. Still, once bitten, twice shy - I did suspect they would turn out to have little to do with weighing the picnic ingredients.

Anyway, I made myself listen to their blandishments, & was entirely swayed by their claim to offer a New & Better Way to show big numbers on graphs. I learned that they were useful tools for digging a graph (hence "plot") & that a graph so dug could be informally addressed as a "log plot", or even - if you only dug it from North to South & left the East-West business to the politicians - as a "semi log plot". (Actually I'm fibbing here - I picked up that last bit on the web, not on T175.)

My comfort was short-lived. This had all been a ploy, to soften me up, a kind of pre-conversion grooming process... suddenly they wanted me to believe that the halfway point between one & ten is not five, but three-&-a-bit. Sign here, cast off your tired old delusions & join us in the alternate reality of the log plot. I resisted, I really did - you have to believe me. I printed off their little boxed explanation. I jeered at it. I cursed it.

But it was too late - I'd been exposed & infected. Halfway through a plaintive email to my Teacher, bemoaning my unfitness to be called to the world of the log plot, a light shone. It was a Wondrous Light. While I was still dazzled, a greater force wrote these words through me:

Halfway between 1 & 10 can't be 5, because you'd have to multiply 1 (the starting point) by 5 to get there, then multiply 5 (the new starting point) by 5 to get from there to 10. That wouldn't work because 5x5 is 25, not 10. So you have to suss out which figure it will work with. Something that you can reach by multiplying 1 by, then get to 10 from by multiplying it by the same number as in the first step.

This makes no sense if you have not been touched by the truth.

Douglas Adams lied. The answer is not 42, it's three-&-a-bit.