Scientific notation and index and log laws

What the exam asks

This is the machinery behind most calculation items in chemistry and physics, and it is examined by being needed rather than by being asked about. Expect to combine two or three quantities in scientific notation under time, and expect pH and pKa items where the log work is the whole question. Two traps do most of the damage. The first is adding numbers with different exponents by adding the exponents, which produces an answer that is wrong by orders of magnitude and does not look wrong. The second is subtracting exponents in the wrong order when dividing, which flips the sign: the exponent of the answer is the top one minus the bottom one, always in that direction. If a division leaves you unsure, check with a case you know, such as 10 to the 6 over 10 to the 2 being 10 to the 4.

Almost all the arithmetic in Section III is one multiplication or one division of numbers written in scientific notation. Concentrations, rate constants, wavelengths, Avogadro's number, dissociation constants: they all arrive as a small number times a power of ten, and they leave the same way. If you can multiply and divide in this notation without thinking, you have removed most of the risk in the paper.

The procedure has three parts and never changes. Handle the mantissas with ordinary arithmetic. Handle the exponents by adding them when you multiply and subtracting them when you divide. Then normalise, which means shifting the decimal point until the mantissa sits between 1 and 10 and paying for each shift with a change in the exponent. Addition and subtraction are the exception: they need both numbers on the same power of ten before you can do anything at all, because you cannot add a thousand to a million by adding 3 and 6.

Index laws and log laws are the same laws seen from two sides. A log is an exponent: log base 10 of x is just the power you raise 10 to in order to get x. Indices add when powers multiply, and that single fact is why logs of a product add. Every log law you need can be recovered from an index law you already believe, which is worth knowing on a morning when a law will not come back to you.

What to hold

• Scientific notation writes a number as a mantissa between 1 and 10 times a power of ten, and normalising at the end means restoring that range.
• Multiplying multiplies the mantissas and adds the exponents; dividing divides the mantissas and subtracts the exponents.
• Adding or subtracting requires both numbers on the same exponent first, because only matched powers of ten can be combined.
• Raising to a power raises the mantissa to that power and multiplies the exponent by it, so a power of a power multiplies the indices rather than adding them.
• Shifting a decimal point one place left raises the exponent by one and one place right lowers it by one, which keeps the value fixed.
• A negative exponent marks a small positive number, not a negative one: 10 to the minus 3 is one thousandth.
• Anything to the power zero is 1, and a fractional index is a root, so x to the power one half is the square root of x.
• To take a square root in scientific notation, first shift the decimal so the exponent is even, then halve the exponent and root the mantissa.
• A log is an exponent: log base 10 of x is the power of ten that gives x, which is why log laws mirror index laws exactly.
• The log of a product is the sum of the logs, the log of a quotient is the difference, and the log of a power brings the power down in front.
• The log of a sum is not the sum of the logs, and there is no law that opens up log of a plus b.
• Worth carrying: log base 10 of 2 is about 0.30 and of 3 is about 0.48, so log of 5 is 1 minus 0.30, which is 0.70.