# A few logarithmic tricks

I *love* a good logarithm. Logarithms are a standby for when I want to work something out ninja-style, and there’s something very satisfying about taking something horrible in the powers, bringing it down to the working line, and finding that it wasn’t so horrible after all.

I’m an old hand, though. You - at least, if you’re in my target audience - may be newer to the wonders of the logarithm. You hopefully have the elementary laws down pat - but perhaps you still struggle to get the answer out.

This article gives some of my best advice for doing that.

### 1. Don’t make it harder than it needs to be

I suppose this is advice for the whole of maths, but I have something specific in mind here. Logarithms are algebraic furniture and can be shuffled around like $x$s or numbers. If you have a logarithmic equation to solve and one of the terms has a minus sign attached, it’s perfectly ok to add that to both sides so it becomes a plus on the other. In fact, I’d recommend that.

Similarly, if you’ve got a fractional multiplier you don’t like the look of? Multiply the whole thing through by it. The equation will remain true.

### 2. Single base

This doesn’t come up often at A-level, but when it does, it’s a bloodbath: if you have logs with different bases, convert them all to the same base. (The formula, in case you’ve forgotten, is $\log_a(b) = \frac{\log_k(b)}{\log_k(a)}$, for whatever valid log base $k$ you like.)

### 3. Get rid of pure numbers

You have a 1, or a 5, or some other ungodly number polluting your lovely joined-up logarithms? It’s not a problem! Simply turn it into a logarithm. Something like 5 is the same as $\log_a(a^5)$, by definition - and now you have something you can fold in to the other logs.

### 4. Factorise

If you’ve got a nasty-looking quadratic in one of your log brackets, it may be worth seeing if you can split it out into two factors - just in case one of them cancels with another log you have lying around. It doesn’t always work - but when it does, it saves you a lot of time and effort.

### 5. Order of operations

Lastly, and probably most importantly: respect the order of operations when you’re solving things. You multiply before you add, so $2\log(x+2) + \log(x)$ is $\log\left((x+2)^2\right) + \log(x)$ rather than $2\log( x(x+2))$. Make sure you work things in the correct order!

I’m sure I’ve missed out some good logarithm advice. If you have any to share, drop it in the comments!