Understanding Logarithms: A Guide for Beginners

August 7th, 2026

Logarithms sound like something from a high-school exam, but they show up constantly in real spreadsheet work: compound growth, order-of-magnitude comparisons, scientific scales, and any time you need to solve for an exponent. Google Sheets gives you three main functions for this — LOG, LOG10, and LN — plus EXP as the inverse of the natural log. This guide explains what they mean in plain language and how to use them.

What a logarithm actually answers

A logarithm answers one question: to what power do I raise this base to get that number?

If 10² = 100, then the base-10 logarithm of 100 is 2. Written as a formula:

=LOG10(100)

That returns 2. Same idea with other bases: 2³ = 8, so the base-2 log of 8 is 3:

=LOG(8, 2)

That returns 3. Once you internalize "logarithm = find the exponent," the rest of the functions are just different defaults for the base.

The three log functions in Google Sheets

LOG — any base (defaults to 10)

LOG takes a value and an optional base. Omit the base and you get a base-10 log, the same as LOG10.

=LOG(100)
=LOG(100, 10)
=LOG(8, 2)

The first two both return 2. The third returns 3. Use an explicit second argument whenever the base matters for the problem — binary (2), natural growth contexts (e), or any custom scale.

LOG10 — always base 10

LOG10 is the specialized form. Prefer it when you mean base 10 and want the formula to read clearly:

=LOG10(1000)

Returns 3, because 10³ = 1000. Base-10 logs are the usual choice for order-of-magnitude work and many scientific scales (decibels, pH, Richter-style magnitudes).

LN — natural logarithm (base e)

LN always uses base e (Euler's number, about 2.718). Natural logs pair with continuous growth and decay models — finance, populations, radioactive half-life style math.

=LN(10)

Returns roughly 2.302585. You rarely need the raw number alone; you almost always combine LN with other values (rates, time periods) as shown below.

EXP undoes LN

EXP raises e to a power. That makes it the inverse of LN:

=EXP(LN(5))
=LN(EXP(5))

Both return 5 (within floating-point precision). In growth formulas you typically use them as a pair: LN to recover a rate from observed values, EXP to project forward from a rate.

Practical patterns

How many years until money doubles?

Solve (1 + rate)^years = 2 for years by taking a log of both sides:

=LOG(2, 1+B2)

With B2 at 0.07 (7% per year), this returns about 10.24 years — the classic "rule of 72" made exact. Any base works here because you are only solving for an exponent; LOG(2, 1+B2) is the clean form because the growth factor itself is the base.

Continuous growth rate from start and end values

If a value grew from A2 to B2 over C2 years under continuous compounding, recover the rate with:

=LN(B2/A2)/C2

Project forward the other way with EXP:

=A2*EXP(0.03*10)

That grows the starting value in A2 at a continuous 3% rate for 10 years.

Order of magnitude (how many digits?)

LOG10 tells you roughly how large a number is on a powers-of-ten scale:

=INT(LOG10(A2))+1

For a positive integer in A2, this returns the number of digits. More generally, LOG10 compresses values that span many orders of magnitude into a range a chart or comparison can actually show — useful for traffic, revenue tiers, sensor readings, and scientific measurements.

Change of base (when you need a base Sheets does not hard-code)

Any log base can be rewritten with natural logs:

=LN(A2)/LN(B2)

That is log of A2 with base B2. Equivalent to =LOG(A2, B2), so you usually prefer LOG — but the identity is useful when building formulas that already work in natural logs.

Common mistakes

• Log of zero or a negative number. LOG(0), LOG(-1), and the same with LN/LOG10 return #NUM!. Logs are only defined for positive values.
• Base must be positive and not 1. =LOG(8, 1) and =LOG(8, -2) are invalid.
• Confusing LOG with LN. In some programming languages log means natural log. In Google Sheets (and Excel), LOG defaults to base 10; use LN for base e.
• Forgetting the inverse pair. If you take LN of a ratio to get a continuous rate, multiply and EXP to apply that rate — mixing discrete (^) and continuous (EXP) models inconsistently will silently skew results.

Which function should you pick?

| Goal | Function | | --- | --- | | Base-10 log, readable formula | LOG10 | | Custom base (2, 10, growth factor, …) | LOG(value, base) | | Continuous growth / natural log | LN | | Undo a natural log / continuous projection | EXP |

Going further

For a broader tour of Sheets' scientific toolkit — trig, logs, and complex numbers together — see Trigonometry, Logarithms, and Complex Numbers in Google Sheets. For the full reference on each function, open the pages for LOG, LOG10, LN, and EXP.