Logarithms as questions about exponents

TL;DR:

• log_b(x) asks: what power of b gives x? So log₂ 32 = 5 because 2⁵ = 32.

• Logarithms are the inverse of exponentials, and their rules come straight from the exponent rules.

• Change of base, log_b x = ln x / ln b, lets any calculator evaluate any base.

Suppose a savings balance doubles every year and you want to know how long until it is 32 times the start. You are asking: how many doublings? That question has a name. It is a logarithm, and the answer to this one is 5.

A logarithm is the exponent

Exponents answer "what do I get if I multiply b by itself n times?" Logarithms turn that around: given the result, find n. The statement log_b(x) = n means exactly bⁿ = x. The two are the same fact in two outfits.

Worked example: log₂ 32 = 5

1. Read the question: "2 to what power gives 32?"

2. Climb the powers of 2: 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32.

3. So the exponent is 5, and log₂ 32 = 5.

Check by going the other way: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Correct.

More short ones to get a feel. log₁₀ 1000 = 3 because 10³ = 1000. log₃ 81 = 4 because 3⁴ = 81. log₂ 1 = 0 because 2⁰ = 1. And log₂(1/8) = −3 because 2⁻³ = 1/8. A logarithm can be negative, but the number inside it cannot be zero or negative, since no power of a positive base lands there.

Why the log rules are the exponent rules in disguise

Exponents add when you multiply: 2³ × 2² = 2⁵. Since logarithms are exponents, the log of a product is the sum of the logs.

• Product. log₂(8 × 4) = log₂ 32 = 5, and log₂ 8 + log₂ 4 = 3 + 2 = 5.

• Quotient. log₂(32 ÷ 4) = log₂ 8 = 3, and log₂ 32 − log₂ 4 = 5 − 2 = 3.

• Power. log₂(8²) = log₂ 64 = 6, and 2 × log₂ 8 = 2 × 3 = 6.

In symbols, log(ab) = log a + log b, log(a/b) = log a − log b and log(aⁿ) = n log a. Before calculators, this was the point: it turned hard multiplication into easier addition. Today the value is that it turns an unknown exponent into something you can bring down and solve.

Change of base: using a calculator for any base

Most calculators have only log (base 10) and ln (base e). To find log₂ 10, use log_b x = ln x / ln b. Here that is ln 10 / ln 2 ≈ 2.3026 / 0.6931 ≈ 3.32.

Check by raising: 2^3.32 = 2³ × 2^0.32 ≈ 8 × 1.25 = 10. It matches. The reason the formula works is short. If y = log_b x, then bʸ = x. Taking ln of both sides gives y ln b = ln x, so y = ln x / ln b.

Solving an equation with the unknown in the exponent

Say a quantity doubles each year from 1 and you want to know when it reaches 50. Solve 2ˣ = 50. Then x = log₂ 50 = ln 50 / ln 2 ≈ 3.912 / 0.693 ≈ 5.64 years.

Does that make sense? In 5 years it is 2⁵ = 32, and in 6 years it is 2⁶ = 64. Fifty lies between, closer to 64, so 5.64 is a sensible answer. Always compare with the nearest whole powers.

Try two simplifications

Combine log₅ 125 + log₅ 5. The product rule gives log₅(125 × 5) = log₅ 625 = 4, since 5⁴ = 625. Check separately: log₅ 125 = 3 and log₅ 5 = 1, and 3 + 1 = 4.

Now combine log₂ 48 − log₂ 3. The quotient rule gives log₂(48 ÷ 3) = log₂ 16 = 4. Check: 48 ÷ 3 = 16 and 2⁴ = 16. Each time, the rule collapsed two logs into one that you could read off as an exponent.

How to check your answer

• Rewrite as an exponent. If you found log₃ 81 = 4, test 3⁴ = 81.

• Bracket it. Find two whole powers on either side of the number, as with 32 and 64 above.

• Check the domain. The input of a log must be positive, and the base must be positive and not 1.

Common mistakes

• Believing log(a + b) = log a + log b. Try a = 2, b = 6 in base 2: log₂ 8 = 3, but log₂ 2 + log₂ 6 = 1 + 2.585 = 3.585. They differ.

• Writing log(a) / log(b) as log(a / b). The first is a ratio of logs, the second is a difference.

• Reading log₂ 32 as 2 × 32 or as 32 divided by 2.

• Forgetting which number is the base and which is the result.

Where you will use it next

Logarithms undo exponential growth, so they give doubling times in exponential growth vs linear growth and solve for the number of years in compound interest step by step. Because they compress huge ranges into small numbers, they also appear in scales for sound, earthquakes and acidity, where each step up multiplies the quantity by a fixed factor. To revisit the rules behind them, see exponent rules and why they follow from counting factors.

Frequently asked questions

What is the difference between log and ln?

log usually means base 10 and ln means base e, about 2.718. Some textbooks use log for base e, so check the book.

Why can't you take the log of a negative number?

A positive base raised to any real power gives a positive result, so no exponent produces a negative number.