Logarithm base conversion:
formula and worked example

What a logarithm in a chosen base answers

A logarithm asks the reverse of a power question. Instead of asking what 2 to the power of 6 is, it asks: what power of 2 gives 64? The answer, 6, is the logarithm of 64 in base 2.

The base matters because different fields count in different units. Computing counts how many times a quantity can be halved, so base 2 is natural: 64 values need 6 binary digits. Everyday digit counting uses base 10, where a number's digit count is one more than the whole part of its log. Growth and decay models in science tend to use base e, the natural log.

Most calculators only have ln and log10 keys, so a conversion rule is needed to reach any other base. That rule is the change-of-base formula.

The change-of-base rule and its inputs

Take the natural log of the number and divide by the natural log of the base. Any common log works equally well, as long as the same one is used on top and bottom.

• Number: the value whose logarithm you want. It must be greater than zero.
• Base: the base you want the log expressed in. It must be positive and not equal to 1.
• Result: the exponent to which the base must be raised to give the number.

Using log10 on both lines gives the same answer as ln on both lines, because the conversion factor cancels in the ratio.

Worked example: the base-2 log of 64

Find how many times 2 must be multiplied by itself to reach 64.

1. 1Natural log of the number: ln 64 ≈ 4.1589.
2. 2Natural log of the base: ln 2 ≈ 0.6931.
3. 3Divide: 4.1589 ÷ 0.6931 ≈ 6.0000, which is exactly 6.

Verify by raising: 2^6 = 64. A conversion result that does not reproduce the number when raised is wrong.

What the converted value means

A whole-number answer, like 6, means the number is an exact power of the base. Most real inputs are not: the base-2 log of 100 is about 6.64, which says 100 lies between 2^6 = 64 and 2^7 = 128, closer to the 7th power.

In computing, the base-2 log, rounded up, gives the number of bits needed to represent that many distinct values: 100 values need 7 bits. In search, it approximates how many halving steps a binary search takes to narrow a sorted list down to one item.

In other settings, base 10 gives order of magnitude. A value with a log10 of 4.3 is roughly 20,000, comfortably between 10^4 and 10^5. When data span huge ranges, taking logs puts them on a scale where equal steps mean equal multiples.

What a percentage change in the number does to the log

Logs turn multiplication into addition. Multiplying the number by 1.1 adds log(1.1) to the result, a fixed amount that does not depend on the starting value. In base 2 that fixed amount is about 0.1375.

This is why a log scale treats a 10% rise from 64 and a 10% rise from 6,400 as the same step. Doubling the number always adds exactly 1 in base 2, which is the usual way to remember the scale.

Domain and arithmetic errors

• Using a base of 1. Every power of 1 is 1, so there is no log, and the formula divides by ln 1 = 0.
• Entering zero or a negative number. Real logarithms are only defined for positive numbers.
• Mixing log types: dividing log10 of the number by ln of the base, which gives a meaningless value.
• Rounding the intermediate logs too early, which can make an exact answer like 6 show up as 5.99 or 6.01.

Checking how many digits or bits a value needs

How many bits are needed to number 1,000 distinct items, and how does the formula get there?

1. 1Compute ln(1000) ÷ ln(2) ≈ 6.9078 ÷ 0.6931 ≈ 9.97.
2. 2Round up to the next whole number: 10.
3. 3Confirm: 2^10 = 1,024, which is at least 1,000, while 2^9 = 512 is not.

Rounding down would leave some items without a code, so the ceiling is the right rounding in this use.