Log(arithm)s Explained

- Ever seen a nice number and thought “I wonder what power of <insert number here> that is”?
- Neither have I, except with awesome powers of two
- People have invented notation for working out what something to the power of something else is, so it’s only logical that they’d invent notation for working out the index from the other two numbers
- Say I wanted to know how many times you had to multiply 2 by itself to get 1024…
- The base is 2, because that’s what I’m trying to figure out the power of
- The target sort of thing is 1024
- I’d do log to the base 2 of 1024 = …
- *calculator screen refreshes*
- The answer is 10, because 2^10=1024
- So, when you see log base something of something else = other something, remember that:
- something to the power of other something equals something else
- That was clear…

Log Rules

- Sometimes, a base isn’t specified. In that case, the base is 10
- Log of 1 (no matter what the base is) is always 0…
- …because anything to the power of zero is one
- If the base is the same as the target thing, the answer is 1…
- …because raising something to the power of 1 does nothing
- Adding two logs of the same base is the same as logging (to the same base) of the targets multiplied
- The same goes for subtraction: they divide
- Making a log negative is the same as logging the reciprocal of the target (as you could work out yourself from the following rule…)
- Where the target is raised to a power, you can put the power in front of the log (it multiplies)
- When you need to change the base of a log…
- …log new base of number = (log old base of number) divided by (log old base of new base)
- Don’t learn these from me. This probably doesn’t make any sense
- Make sure you learn these from somewhere, though – the rules aren’t given to you in the exam…

Exponential Equations

- When you’re asked to solve for x, an equation where x is the power/index/exponent/whatever-you-call-it of something…
- …such as 7^x = 3
- Log both sides, use the power-to-multiplication rule, then divide
- So x log 7 = log 3, meaning x = log 3 divided by log 7
- Then there are the more complicated exponential equations, which require you to let a letter, say… y, represent base^x
- You then transform the equation into a quadratic, using the fact that base^(2y) = (base^y)^2 [because of the rules of indeces: pre-Core 2 stuff]
- Once you have a quadratic, solve it. Don’t forget to convert y back to base^x at the end, though!

Exponential Graphs

- Graphs of “y equals base to the power of x” are graphs of exponential functions
- The basic y = a^x, when it hasn’t been translated, stretched or otherwise transformed, will cross the y axis at (0, 1)
- It has no x intercept, because although it gets closer and closer to the x-axis, it never actually meets it. Thus, the x-axis is asymptotic to the curve
- I suppose you could have an x-intercept if you transformed the curve to move down a bit, though (I could be wrong…)
- It looks like a curve which starts off gradually, and gets steeper and steeper
- The bigger the base, the steeper the line (well, it gets steeper more quickly)
- If you’re asked to sketch a line like this, you only need to worry about getting the correct shape and intercept
- Where the base is less than 0, different stuff happens. I don’t think you need to worry about that for Core 2, though…

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