CHEM10003 Maths Cheat Sheet PDF

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CHEM10003 — MATHS CHEAT SHEET DR LARS GOERIGK

1. Exponent rules • a1 = a (It does not matter what the base a is.) • Zero-exponent rule: a0 = 1 (It does not matter what the base a is.) • Negative-exponent rule: a−b = a1b √ 1 • ab = b a  c • Power rule: ab = ab·c

• Product rule: ab · ac = ab+c b • Quotient rule: aac = ab−c • (a · m)b = ab · mb , where m is different from a.  a b b • ma b = m

2. Logarithm rules Imagine we have a base a and we raise it to the power b to obtain the result x: ab = x. Then, the “logarithm of x base a” gives us the exponent b: logax = b. A common base is Euler’s number e. Then, loge x is written as ln(x), which we call “natural logarithm”. From the exponent rules, we can work out the following rules for logarithms: • logax = b > 0 if x > 1. • loga1 = 0 (Compare this to the zero-exponent rule above. It does not matter what the base a is.) • logax = b < 0 if 0 < x < 1. (Compare this to the negative-exponent rule above.) • Logarithms of x with x ≤ 0 are not defined because if you take a base a to power b, the result x is always larger than 0. • logaa = 1 • logax + logay = loga(x  ·y) (Only works if both bases are the same.) • logax − logay = loga yx (Only works if both bases are the same.) √ • loga (xc ) = c · logax, which also means that loga ( c x) = logca x . • logax = log1x a • logax =

logm x logm a ,

where m is different from a....