The Laws of Logarithms

Every logarithm is an exponent

A logarithm is a power. ${\log }_{a}x$logₐ x is the power to which a must be raised to give x, so every fact about logarithms is a fact about exponents, read backward. The laws of logarithms come from the laws of exponents in exactly this way.

The law of exponents behind the first law of logarithms is about multiplying powers of the same base. $2^3\times 2^4$2³ × 2⁴ means (2 × 2 × 2) × (2 × 2 × 2 × 2), which is seven 2s multiplied: $2^7$2⁷. Multiplying powers of one base adds the exponents, 3 + 4 = 7.

The product law

Read that fact in logarithms to base 2. $8=2^3$8 = 2³, $16=2^4$16 = 2⁴ and $8\times 16=128=2^7$8 × 16 = 128 = 2⁷. So ${\log }_{2}8=3$log₂ 8 = 3, ${\log }_{2}16=4$log₂ 16 = 4 and ${\log }_{2}128=7$log₂ 128 = 7, and 3 + 4 = 7. The logarithm of the product is the sum of the logarithms.

The same argument works for any positive numbers. Take the common logarithm, to base 10, and write a and b as powers of 10: a = 10ᵐ and $b=10^n$b = 10ⁿ, so that log a = m and log b = n. Then $ab=10^m\times 10^n=10^{m+n}$ab = 10ᵐ × 10ⁿ = 10ᵐ⁺ⁿ, and so log(ab) = m + n = log a + log b.

For example, log 2 = 0.30103 and log 3 = 0.47712, to 5 decimal places, and log 6 = log(2 × 3) = 0.30103 + 0.47712 = 0.77815, which is what a calculator gives for log 6. The argument used base 10, but any base gives the same law, as long as every logarithm in it has the same base.

The top row is x, the powers of 2 from 1 to 128, and the row under it is ${\log }_{2}x$log₂ x. Multiplying two numbers in the top row adds their logarithms: 8 × 16 = 128, and 3 + 4 = 7.

The quotient law

Dividing powers of one base subtracts the exponents: $10^m/10^n=10^{m-n}$10ᵐ / 10ⁿ = 10ᵐ⁻ⁿ. With a = 10ᵐ and $b=10^n$b = 10ⁿ again, $a/b=10^{m-n}$a / b = 10ᵐ⁻ⁿ, so $\log \left(a/b\right)=m-n=\log a-\log b$log(a / b) = m − n = log a − log b. Dividing inside the logarithm becomes subtracting outside it.

In base 2: 128 ÷ 8 = 16, and ${\log }_{2}128-{\log }_{2}8=7-3=4={\log }_{2}16$log₂ 128 − log₂ 8 = 7 − 3 = 4 = log₂ 16. In base 10: $\log 20-\log 4=\log \left(20/4\right)=\log 5$log 20 − log 4 = log(20 / 4) = log 5.

The power law

A power is repeated multiplying, so the product law gives the power law at once for a whole number n. $\log \left(x^3\right)=\log \left(x\times x\times x\right)=\log x+\log x+\log x=3\log x$log(x³) = log(x × x × x) = log x + log x + log x = 3 log x. In general, $\log \left(x^n\right)=n\log x$log(xⁿ) = n log x: the exponent comes to the front as a multiplier.

The law of exponents behind it is the power of a power, which multiplies the exponents. Write x = 10ᵐ, so log x = m. Then $x^n={\left(10^m\right)}^n=10^{mn}$xⁿ = (10ᵐ)ⁿ = 10ᵐⁿ, and $\log \left(x^n\right)=$log(xⁿ) = mn = n log x. This argument does not need n to be a whole number, so the power law holds for every n. For n = ½ it says $\log \sqrt{x}=\frac{1}{2}\log x$log √x = ½ log x, and for n = −1 it says $\log \left(\frac{1}{x}\right)=-\log x$log(1/x) = −log x.

For example, $\log 9=\log 3^2=2\log 3=2\times 0.47712=0.95424$log 9 = log 3² = 2 log 3 = 2 × 0.47712 = 0.95424, and $\log 1000=\log 10^3=3\log 10=3$log 1000 = log 10³ = 3 log 10 = 3.

Two values that are the same in every base

$a^0=1$a⁰ = 1 for every base, so ${\log }_{a}1=0$logₐ 1 = 0. $a^1=a$a¹ = a, so ${\log }_{a}a=1$logₐ a = 1. These fit the laws: log(x × 1) = log x + log 1 = log x + 0, as it must be.

The laws together

The laws can combine several logarithms into one. For 2 log 3 + log 4 − log 6, first use the power law, $2\log 3=\log 3^2=\log 9$2 log 3 = log 3² = log 9. Then the product law gives log 9 + log 4 = log 36, and the quotient law gives $\log 36-\log 6=\log \left(36/6\right)=\log 6$log 36 − log 6 = log(36 / 6) = log 6.

They can also split one logarithm into several. $\log \left(x^{2}y/z\right)=\log x^2+\log y-\log z=2\log x+\log y-\log z$log(x²y / z) = log x² + log y − log z = 2 log x + log y − log z.

What the laws do not say

The laws turn multiplying into adding. They say nothing about the logarithm of a sum, and log(a + b) is not log a + log b. Test it with numbers: log(10 + 10) = log 20 = 1.3010, but log 10 + log 10 = 1 + 1 = 2, which is log 100, the logarithm of 10 × 10.

Nor is the product of two logarithms the logarithm of the product. log 100 × log 10 = 2 × 1 = 2, but log(100 × 10) = log 1000 = 3. And a quotient of logarithms is not the logarithm of the quotient: $\log 100/\log 10=2/1=2$log 100 / log 10 = 2 / 1 = 2, while $\log \left(100/10\right)=\log 10=1$log(100 / 10) = log 10 = 1.

The power law brings the exponent of x to the front. It does not apply to a power of the whole logarithm: ${\left(\log x\right)}^2$(log x)² means log x × log x, which is not 2 log x. At x = 1000, ${\left(\log 1000\right)}^2=3^2=9$(log 1000)² = 3² = 9, while 2 log 1000 = 2 × 3 = 6.

Three pairs of numbers a and b: 2 and 5, 10 and 10, 4 and 25, with common logarithms to 4 decimal places. The first two columns agree every time; the third, the logarithm of the sum, never does.

The usual mistakes

Adding inside the logarithm. log 4 + log 5 is log(4 × 5) = log 20, not log(4 + 5) = log 9.

Subtracting inside the logarithm. log 20 − log 4 is $\log \left(20/4\right)=\log 5$log(20 / 4) = log 5, not log 16.

Cubing the logarithm. $\log \left(x^3\right)$log(x³) is 3 log x, not ${\left(\log x\right)}^3$(log x)³: the power sits on the x inside, and it comes down as a multiplier.

Mixing bases. ${\log }_{2}8+{\log }_{10}10=3+1=4$log₂ 8 + log₁₀ 10 = 3 + 1 = 4, but ${\log }_{2}80$log₂ 80 is about 6.32 and ${\log }_{10}80$log₁₀ 80 is about 1.90, and neither is 4. The laws hold only when every logarithm has the same base.

A power law as a straight line

The next problem fits a model of the form $B=a\times M^n$B = a × Mⁿ to measured data. Take the logarithm of both sides and use two laws at once. The product law splits the right side: $\log B=\log a+\log M^n$log B = log a + log Mⁿ. The power law brings n to the front: log B = log a + n log M.

Write X for log M and Y for log B. The equation becomes Y = nX + log a, which is a straight line, with gradient n and intercept log a. So if the points with coordinates log M and log B lie on a straight line, the data follow a power law: the gradient of the line is the power n, and the intercept is log a, so a is 10 to the power of the intercept.