The laws of logarithms are a set of rules that let you break apart, combine, or rearrange logarithmic expressions. There are three core laws (product, quotient, and power), plus a handful of foundational properties and a change-of-base formula. Together, they turn complicated expressions into simpler ones and make logarithms practical across math, science, and engineering.
To make sense of these laws, it helps to understand what a logarithm actually is. A logarithm answers the question: “What exponent do I need?” If you know that 2 raised to the 3rd power equals 8, then the logarithm base 2 of 8 is 3. In notation, that’s log₂(8) = 3. The logarithm is the inverse of an exponential function. If y = bˣ, then x = log_b(y). This inverse relationship is the foundation everything else builds on.
Two Properties Worth Memorizing
Before diving into the three main laws, two simple properties come up constantly:
- log_b(1) = 0 for any valid base. This is because any number raised to the power of zero equals 1.
- log_b(b) = 1 for any valid base. The base raised to the first power is just itself.
These aren’t complicated, but they simplify expressions faster than you’d expect. If you’re working through a problem and spot log₅(1), you can replace it with 0 immediately. See log₇(7), and it’s just 1.
The Product Rule
The product rule says that the logarithm of a product equals the sum of the individual logarithms:
log_b(M × N) = log_b(M) + log_b(N)
So instead of finding the log of a large product, you can split it into smaller, more manageable pieces. For example, log₂(32) can be rewritten as log₂(8 × 4), which becomes log₂(8) + log₂(4) = 3 + 2 = 5.
This rule extends to any number of factors. If you have four values multiplied together, you get four separate logarithms added together: log_b(w × x × y × z) = log_b(w) + log_b(x) + log_b(y) + log_b(z). The product rule is why logarithms were historically used to simplify multiplication. Before calculators, multiplying large numbers meant looking up their logarithms in a table, adding those values, and converting back.
The Quotient Rule
The quotient rule is the mirror image of the product rule. The logarithm of a fraction equals the difference of two logarithms:
log_b(M / N) = log_b(M) − log_b(N)
You subtract the logarithm of the denominator from the logarithm of the numerator. For example, log₃(81/9) = log₃(81) − log₃(9) = 4 − 2 = 2. And you can verify: 3² = 9, while 81/9 = 9. It checks out.
When using this rule, simplify the fraction first if the numerator and denominator share common factors. Canceling before applying the rule keeps the arithmetic cleaner.
The Power Rule
The power rule lets you pull an exponent out of a logarithm and turn it into a multiplier:
log_b(Mⁿ) = n × log_b(M)
This is arguably the most useful of the three laws, because exponents inside logarithms are common and awkward to work with. The power rule makes them straightforward. For instance, log₂(8⁴) becomes 4 × log₂(8) = 4 × 3 = 12.
You can see why this works by thinking of the product rule. Take log_b(x²). That’s the same as log_b(x × x), which the product rule expands to log_b(x) + log_b(x), which is simply 2 × log_b(x). The same logic applies to any exponent. The power rule also works with fractional exponents, which means you can use it to handle square roots and cube roots. Since √x = x^(1/2), the logarithm of a square root becomes (1/2) × log_b(x).
The Change of Base Formula
Most calculators only have buttons for two types of logarithms: base 10 (log) and base e (ln). If you need a logarithm with a different base, the change of base formula converts it:
log_a(x) = log_b(x) / log_b(a)
You pick any base b you want for the conversion, as long as you use the same base in both the numerator and the denominator. In practice, people almost always convert to base 10 or base e because those are what’s available on a calculator. So log₅(20) becomes log(20) / log(5), which you can punch into any standard calculator to get approximately 1.861.
Natural and Common Logarithms
Two bases show up far more often than any others. The common logarithm uses base 10 and is written simply as “log” with no subscript. The natural logarithm uses base e (approximately 2.71828) and is written as “ln.”
The natural logarithm follows all the same laws: ln(a × b) = ln(a) + ln(b), ln(a/b) = ln(a) − ln(b), and ln(aⁿ) = n × ln(a). It appears throughout calculus and physics because it has a uniquely clean derivative: the derivative of ln(x) is simply 1/x. That mathematical elegance is why e became the default base in higher math and most scientific fields.
Common logarithms (base 10) tend to dominate in applied measurement scales, which brings us to where these laws actually show up in real life.
Where Logarithmic Laws Show Up
Logarithms aren’t just abstract algebra. Several widely used measurement systems are built on them.
The pH scale in chemistry is defined as pH = −log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration of a solution. Because the scale is logarithmic, a change of one pH unit corresponds to a tenfold change in acidity. That’s the power rule and the properties of base-10 logarithms at work. A solution with pH 3 is ten times more acidic than one with pH 4, and a hundred times more acidic than pH 5.
Sound intensity is measured in decibels using the formula dB = 10 × log₁₀(I/I₀), where I₀ is the threshold of human hearing. The logarithmic scale reflects how your ears actually perceive volume. Doubling the physical intensity of a sound doesn’t make it sound twice as loud; it adds roughly 3 decibels. Without the quotient rule built into this formula, comparing two sound sources would require working with unwieldy ratios of raw intensity values.
Earthquake magnitude works similarly. The Richter scale is determined from the logarithm of the amplitude of seismic waves recorded by seismographs. The moment magnitude scale, which largely replaced it, uses the formula M_W = (2/3)(log₁₀(M₀) − 9.1), where M₀ is the seismic moment. Each whole number increase in magnitude represents roughly a 31.6-fold increase in energy released.
Domain Restrictions to Keep in Mind
One important constraint applies to every logarithm: you can only take the logarithm of a positive number. The domain of log_b(x) is x > 0. There is no real-number answer for log_b(0) or for the logarithm of any negative number. The base b also has restrictions: it must be positive and cannot equal 1.
This matters when you’re applying the laws. If you’re using the quotient rule to expand log(M/N), both M and N must be positive. If you’re using the power rule on log(Mⁿ), the result of Mⁿ must be positive. Ignoring these restrictions is one of the most common sources of errors when simplifying logarithmic expressions.