The Change of Base Rule

A logarithm the calculator does not have

${\log }_{2}20$log₂ 20 asks: what power of 2 gives 20? Since $2^4=16$2⁴ = 16 and $2^5=32$2⁵ = 32, the answer is between 4 and 5, but no whole number works, and there is no ${\log }_2$log₂ key to press.

A calculator has two logarithm keys. log is the common logarithm, to base 10. ln is the natural logarithm, whose base is a number called e, about 2.718. The change of base rule turns a logarithm in any base into a quotient of logarithms in one of those two bases.

Deriving the rule

Call the unknown logarithm x, so $x={\log }_{2}20$x = log₂ 20. By the meaning of a logarithm, that says $2^x=20$2ˣ = 20. The unknown is now an exponent.

Take the common logarithm of both sides: $\log \left(2^x\right)=\log 20$log(2ˣ) = log 20. The power law of logarithms says that the logarithm of a power is the exponent times the logarithm, so the exponent comes down to the front: x log 2 = log 20. Now x is no longer an exponent, and dividing both sides by log 2 gives $x=\log 20/\log 2$x = log 20 / log 2.

But x was ${\log }_{2}20$log₂ 20 all along. So ${\log }_{2}20=\log 20/\log 2=1.3010/0.3010=4.322$log₂ 20 = log 20 / log 2 = 1.3010 / 0.3010 = 4.322, to 3 decimal places. It lies between 4 and 5, as it should.

Any base at all

Nothing in that argument depended on 2 or on 20. For any base a, a positive number other than 1, and any positive x, write $y={\log }_{a}x$y = logₐ x, so $a^y=x$aʸ = x. Taking common logarithms gives y log a = log x, so $y=\log x/\log a$y = log x / log a. That is the change of base rule: ${\log }_{a}x=\log x/\log a$logₐ x = log x / log a.

The logarithm of the number goes on top, and the logarithm of the base goes underneath. Any one base will do on the right, as long as the top and the bottom use the same base: ${\log }_{a}x=\ln x/\ln a$logₐ x = ln x / ln a gives exactly the same answer. For ${\log }_{2}20$log₂ 20 that is 2.9957 ÷ 0.6931 = 4.322 again.

Using it, and checking

Find ${\log }_{5}30$log₅ 30. By the rule it is $\log 30/\log 5$log 30 / log 5. A calculator gives log 30 = 1.4771 and log 5 = 0.6990, to 4 decimal places, so ${\log }_{5}30=1.4771/0.6990=2.113$log₅ 30 = 1.4771 / 0.6990 = 2.113, to 3 decimal places.

A logarithm is a power, so the answer can be put back to check it: $5^{2.113}=29.99$5^2.113 = 29.99, to 2 decimal places, which is 30 to the accuracy of the rounded logarithms. The answer is also sensible: $5^2=25$5² = 25 and $5^3=125$5³ = 125, and 30 is just above 25, so the power is just above 2.

In the same way, ${\log }_{3}7=\log 7/\log 3=0.8451/0.4771=1.771$log₃ 7 = log 7 / log 3 = 0.8451 / 0.4771 = 1.771, to 3 decimal places.

Exact answers

The rule works with any base on the right, so choose a base that makes the numbers easy. ${\log }_{4}8={\log }_{2}8/{\log }_{2}4=3/2=\frac{3}{2}$log₄ 8 = log₂ 8 / log₂ 4 = 3 / 2 = 3/2. Check: $4^{\frac{3}{2}}$4^(3/2) is the square root of 4, cubed, which is $2^3=8$2³ = 8.

In the same way ${\log }_{8}2={\log }_{2}2/{\log }_{2}8=\frac{1}{3}$log₈ 2 = log₂ 2 / log₂ 8 = 1/3, because 2 is the cube root of 8.

The rule as a stretch

The rule says ${\log }_{2}x=\log x\times \left(1/\log 2\right)$log₂ x = log x × (1 / log 2), and $1/\log 2=3.32$1 / log 2 = 3.32, to 2 decimal places. That number is the same for every x. So every value of ${\log }_{2}x$log₂ x is the value of log x multiplied by the same 3.32, and the graph of $y={\log }_{2}x$y = log₂ x is the graph of y = log x stretched vertically by a factor of about 3.32. At x = 8, log 8 = 0.903 and ${\log }_{2}8=3$log₂ 8 = 3, and 3 ÷ 0.903 = 3.32.

The same is true for every base: the graphs of all logarithms are vertical stretches of one another. All of them pass through (1, 0), because the logarithm of 1 is 0 in every base, and a stretch leaves 0 where it is.

The gold curve is $y={\log }_{2}x$y = log₂ x and the white curve is y = log x. Both pass through (1, 0). At x = 8 the gold curve is at 3 and the white curve at 0.903: every height of the gold curve is about 3.32 times the height of the white one.

The usual mistakes

Turning the quotient upside down. ${\log }_{2}40$log₂ 40 is $\log 40/\log 2=5.32$log 40 / log 2 = 5.32, to 2 decimal places. $\log 2/\log 40=0.19$log 2 / log 40 = 0.19 is the logarithm of 2 to base 40, a different number. The base always goes underneath.

Subtracting instead of dividing. log 40 − log 2 is $\log \left(40/2\right)=\log 20=1.30$log(40 / 2) = log 20 = 1.30, by the quotient law. It is not a change of base.

Dividing by the base instead of its logarithm. $\left(\log 40\right)/2=0.80$(log 40) / 2 = 0.80 is not ${\log }_{2}40$log₂ 40. Both the number and the base need their logarithms taken.