Two bases are used often enough to get their own notation: log x means base 10 and ln x means base e. Neither is a new function. The skill is to expose the hidden base, use a matching inverse exactly when one is present, and bracket any decimal before a calculator rounds it.
Core ideas
Neither notation adds a rule. Each names a base that comes up often enough to be worth an abbreviation.
Written with no base at all, log is base 10 in this course, on a calculator key, and in school algebra generally. In an analysis text it usually means base e; in a complexity bound it usually means base 2. The symbol does not carry its base, the subject does, and the convention is arbitrary.
ln x means log_e(x), with e = 2.71828… produced in “The Number e” by compounding faster and faster. It removes an e-power in one inverse step: ln(e^u) = u. Any legal logarithm could be applied to an equation containing e^u; ln is the matched and simplest choice, not the only legal one.
10^3 < 4500 < 10^4 puts log 4500 between 3 and 4. Likewise e² < 10 < e³ puts ln 10 between 2 and 3. A bracket catches the wrong base, a dropped sign and an implausible decimal before the keys can make the mistake look precise.
The material
The definitions, the inverse identities they inherit, and how to choose and check the useful one.
log x, written with no base, means log_10(x): the exponent to which 10 must be raised to give x. Its domain is x > 0, as for every logarithm.
ln x means log_e(x), where e = 2.71828… is the number “The Number e” obtained as a limit. The name is historical — logarithmus naturalis — and e is not a more natural base for arithmetic than any other. It is the base in which continuous growth is written, which is the only reason this course needs it.
Nothing about the two definitions is new. Every statement from “What a Logarithm Is” onwards applies to them unchanged, because they are instances of the same definition and not a second kind of object.
The two bottom lines are the same fact in the two possible orders, and they carry different conditions for a reason. log(10^k) = k holds for every real k because 10^k is positive whatever k is. 10^(log x) = x needs x > 0 first, because otherwise log x does not exist to be exponentiated.
When the power and the logarithm use the same base, they undo one another exactly: log(10^k) = k and ln(e^k) = k. Cross the bases and there is no cancellation: log(e^k) = k·log e and ln(10^k) = k·ln 10. The expressions are still exact, but a constant scale factor remains.
From e^x = 12, taking ln gives x = ln 12 in one step. Taking a common logarithm is also legal: x·log e = log 12, so x = log 12 / log e. Change of base in “Change of Base” proves those exact expressions are equal. ln is the shorter route because its base matches the power, not because every other logarithm is forbidden.
The base supplies a free bracket. Since 10^3 < 4500 < 10^4, 3 < log 4500 < 4. Since e² < 10 < e³, 2 < ln 10 < 3. A decimal outside the right interval used the wrong base, lost a sign or was entered incorrectly. The bracket does not compute the answer; it checks what a calculator returns.
For a common logarithm, scientific notation makes that bracket visible: if x = a·10^k with 1 ≤ a < 10, then log x = k + log a and k ≤ log x < k + 1. The attached lab shows this exact power-of-ten part beside the rounded remainder as an estimation aid. No table terminology or separate algorithm is needed.
For every x > 0, ln x = ln 10 · log x, and ln 10 ≈ 2.302585.
So the natural logarithm of a number is always about 2.3 times its common logarithm: log 500 ≈ 2.6990 and ln 500 ≈ 6.2146. “Change of Base” shows that every pair of bases is related by a constant of exactly this kind, and derives it in two lines.
That proportionality is also the practical reason the two keys are enough. A standard scientific calculator offers log and ln; “Change of Base” computes every other base from either one.
Interactive
Two exponents for one argument, checked by a power-of-ten bracket
Method
Four decisions: expose the base, check the domain, look for exact cancellation, then estimate.
Rewrite log x as log_10(x) and ln x as log_e(x). The symbol alone does not carry a universal convention, so state the course or context when a bare log could be ambiguous.
Both functions require a positive argument. ln(e^u) is always legal because e^u > 0; e^(ln x) requires x > 0 before ln x exists.
Use log(10^k) = k or ln(e^k) = k exactly. If the bases do not match, keep the scale factor: log(e^k) = k·log e, not k.
Compare the argument with nearby powers of the actual base. Then compute if a decimal is required, carry the exact logarithmic expression through later work, and state the number of places rounded.
First expose the base. Exact inverse pairs cancel; crossed bases keep a factor; an unmatched number is bracketed before it is rounded.
The first two lines cancel because the bases match. The next two do not: log asks for a power of 10, not a power of e, and ln asks for a power of e, not a power of 10. Leaving log e and ln 10 visible is the exact answer, not unfinished work.
The bracket is a check on the decimal rather than a way to manufacture it. An answer such as 1.000000 for ln 10 would reveal a false cancellation immediately, while the calculator would otherwise print that wrong input with perfect confidence.
For a faded named-base pass, annotate the bases in log(10^−6), ln(e^(5/2)) and log(e³). The first move is supplied: write each bare name as log_10 or log_e. Evaluate the matching pairs exactly, leave the crossed pair in an exact form, and bracket it using nearby powers of 10. Check each result by reading it as an exponent.
Given log(10^u), ln(e^u), log(e^u) and ln(10^u), you should cancel only the two matched pairs, keep the scale factor on the crossed pairs, and bracket any requested decimal between powers of the correct base before computing it.
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