MHF4U Laws of Logarithms: Notes and Practice

Logarithms turn multiplication into addition. That sounds like a party trick, but it is the reason logarithms were invented in the first place, and it is why they are the right tool for measuring things that span enormous ranges — earthquake energy, sound intensity, acidity. Every one of the laws below is just an exponent law wearing different clothes, so you can always rebuild them from scratch if you forget one.

Fig 1Each logarithm law is the mirror image of an exponent law: multiply becomes add, divide becomes subtract, power becomes multiply.

A white slide rule with a sliding middle strip and a clear cursor window, its scales covered in numbers and fine tick marks, on a white background. — PhotoBefore electronic calculators, people multiplied with slide rules like this one. The main scales are spaced by logarithms, so sliding one length along another adds two logarithms, and the product law turns that sum into a product.Jacek Halicki, CC BY-SA 4.0, via Wikimedia Commons

The three laws

Law	Rule	Exponent law behind it
Product	$lo g_{b} (mn) = lo g_{b} m + lo g_{b} n$	$b^{p} \cdot b^{q} = b^{p + q}$
Quotient	$lo g_{b} (\frac{m}{n}) = lo g_{b} m - lo g_{b} n$	$\frac{b ^{p}}{b ^{q}} = b^{p - q}$
Power	$lo g_{b} (m^{n}) = n lo g_{b} m$	$(b^{p})^{n} = b^{p n}$

All three require the same base b throughout, and every argument of a logarithm must be positive (m and n in the product and quotient laws, m in the power law); the exponent n in the power law can be any real number.

VideoJensenMath, an Ontario teacher, works through the product, quotient and power laws of logarithms with examples.Watch on YouTube

Where the product law comes from

Let $p = lo g_{b} m$ and $q = lo g_{b} n$ . By the definition, that means $m = b^{p}$ and $n = b^{q}$ . Multiply them together and use the product law for exponents.

mn = b^{p} \cdot b^{q} = b^{p + q} ⟹ lo g_{b} (mn) = p + q = lo g_{b} m + lo g_{b} n

The quotient law comes out of the identical argument with division, and the power law comes from $m^{n} = (b^{p})^{n} = b^{p n}$ , so $lo g_{b} (m^{n}) = p n = n lo g_{b} m$ . Notice how the power law is the most useful of the three in practice: it takes an exponent, which is stuck up high where you cannot reach it, and drags it down into a plain multiplier that you can divide by.

Fig 2The exponent laws you already know — each one produces a logarithm law when read backwards.

Expanding and condensing

Expanding means breaking one logarithm of a complicated expression into a sum and difference of simpler logarithms. Condensing is the reverse. When expanding, work from the outside in: deal with the overall division first, then the multiplications, then the exponents. When condensing, always apply the power law first so that every term has a coefficient of 1, and only then combine with the product and quotient laws.

Rewrite every root as a fractional exponent: $y = y^{1/2}$ , $3 z = z^{1/3}$ .
Use the power law to move exponents out front (expanding) or back inside (condensing).
Terms being added combine as a product; terms being subtracted combine as a quotient.
Evaluate any logarithm that comes out to a clean number, such as $lo g 100 = 2$ , instead of leaving it unsimplified.

Change of base

A calculator only has buttons for base 10 and base $e$ . To evaluate something like $lo g_{7} 100$ , rewrite it in terms of a base the calculator does have. The change of base formula lets you convert a logarithm to any new base $a$ you like. Here is why it works: let $y = lo g_{b} x$ , so $b^{y} = x$ ; take $lo g_{a}$ of both sides to get $lo g_{a} (b^{y}) = lo g_{a} x$ ; by the power law the left side becomes $y lo g_{a} b$ , so $y = \frac{lo g _{a} x}{lo g _{a} b}$ . Change of base is also how you compare logarithms with different bases, since the three laws never let you combine $lo g_{2} 5$ with $lo g_{3} 5$ directly.

lo g_{b} x = \frac{lo g _{a} x}{lo g _{a} b} most often used as lo g_{b} x = \frac{lo g x}{lo g b}

There is no law for a sum inside a logarithm

The product law applies to

lo g_{b} (mn)

, a logarithm of a product. It does not apply to

lo g_{b} (m + n)

, a logarithm of a sum — that expression cannot be broken apart at all. Check it with numbers once and you will never do it again:

lo g_{2} (4 + 4) = lo g_{2} 8 = 3

, but

lo g_{2} 4 + lo g_{2} 4 = 2 + 2 = 4

, and three does not equal four. The same test kills the other invented laws:

\frac{lo g 100}{lo g 10} = \frac{2}{1} = 2

, whereas

lo g 100 - lo g 10 = 2 - 1 = 1

, so a quotient of logarithms is not a difference of logarithms.

Laws of Logarithms

Key things to remember

The three laws

Where the product law comes from

Expanding and condensing

Change of base

Worked examples

Where marks get lost

Flashcards 8

Practice