Maths › Exponentials and logarithms › Logarithms and their laws
Logarithms and their laws
A logarithm answers one question: what power produced this number? That single idea undoes every exponential, drags unknowns down from exponents where algebra cannot otherwise reach them, and obeys three laws that turn multiplication into addition.
Builds on Exponential functions and e.
IN THIS TOPIC
- Convert between exponential and logarithmic statements, in any base and in base e.
- Use the three log laws to combine, split and simplify expressions.
- Solve equations with the unknown in the exponent, including ax = b.
WHAT YOU PROBABLY THINK
log (x + y) = log x + log y.
The inverse of an exponential
The statement an = x and the statement loga x = n say the same thing; a logarithm is an exponent, read backwards. So log10 1000 = 3 because 103 = 1000, and loga a = 1 for any base. The base that matters most is e, whose logarithm gets its own name, ln x, and undoes last lesson's star function.
WORKED EXAMPLE
Undoing e, twice over
Solve e2x+1 = 10, and then ln (3x + 2) = 2, each to 4 significant figures.
For the first, take ln of both sides: 2x + 1 = ln 10, so x = (ln 10 − 1)/2 = 0.6513.
For the second, apply e to both sides: 3x + 2 = e2, so x = (e2 − 2)/3 = 1.796.
One equation needed the log peeled off, the other needed it applied. Deciding which of the pair you are holding is most of the skill.
The three laws
Because logs are exponents, the index laws translate into log laws, all on the must-learn list,
with the third holding for negative and fractional k too, so −log x = log (1/x) and ½ log x = log √x. The laws are why logs exist. A logarithm counts multiplicative steps, and counting is additive. Note what is absent: nothing on the list touches log (x + y), and the opening lie has no law to stand on. A one-line counter example in base 10, log 20 = 1.301 while log 10 + log 10 = 2, finishes it in proof-unit style.
WORKED EXAMPLE
Collapsing to a single logarithm
Write 2 log 3 + log 5 − log 15 as a single logarithm.
The multiple moves inside first: 2 log 3 = log 9.
Then combine left to right: log 9 + log 5 = log 45, and log 45 − log 15 = log (45/15).
The expression is log 3.
Also worth a check the other way: log 9 + log 5 − log 15 must equal log 3 numerically, and it does, 0.9542 + 0.6990 − 1.1761 = 0.4771.
Unknowns in the exponent
When the unknown sits in an exponent and the bases refuse to match, take logs of both sides; the third law then pulls the unknown down as a multiplier, and ordinary algebra finishes.
WORKED EXAMPLE
A mismatched base
Solve 23x−1 = 3, giving the answer to 4 significant figures.
Take logs of both sides: (3x − 1) log 2 = log 3.
So 3x − 1 = log 3/log 2 = 1.585, and x = 2.585/3 = 0.8617.
Any base of logarithm works, log or ln alike, because the base cancels in the ratio. The one wrong move is trying to force 3 into a power of 2 by inspection.
YOUR TURN
Solve for the exponent
Solve 5x = 30, to 4 significant figures, before opening the working.
Show the working
Take logs: x log 5 = log 30, so x = log 30/log 5 = 2.113.
A sense check brackets the answer: 52 = 25 and 53 = 125, so x had to sit just above 2.
TRY IT UNSEEN
Logs against a growth model
Last lesson's population model was P = 500e0.2t. Find, to 3 significant figures, the time at which P reaches 10 000.
Show the working
Set 500e0.2t = 10 000, so e0.2t = 20.
Take ln: 0.2t = ln 20, so t = 5 ln 20 = 15.0 days.
Logs are how every “when does the model reach…” question is answered from now on; the exponential poses the question and its inverse retrieves the time.
THE EXAM BIT
- Translate between forms before doing algebra: an = x and loga x = n are the same fact, and writing the conversion earns the setup mark.
- Quote the log law you use at each step; combined or split logs without a named law lose method marks.
- There is no law for log (x + y). If an addition appears inside a log, stop and look for a factorisation instead.
- For ax = b, take logs in any base and divide; keep the exact ratio until the final rounding.
- ln undoes e and e undoes ln, but only when applied to the whole side of an equation, not term by term.
CHECK YOURSELF
Express log 4 + log 25 as a single logarithm and evaluate it, then solve e3x = 40 to 3 significant figures.
Show a hint
4 × 25 is friendly; then ln both sides.
Show the answer
log 4 + log 25 = log 100 = 2, in base 10.
Taking ln of e3x = 40 gives 3x = ln 40, so x = ln 40/3 = 1.23.
Both answers came from the same idea, a logarithm is the exponent that was used, recovered.
A logarithm is an exponent: a to the n equals x and log base a of x equals n are one statement.
Logs turn multiplication into addition and bring exponents down as multipliers.
CHECK YOUR PROGRESS
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- Convert between exponential and logarithmic statements, in any base and in base e.
- Use the three log laws to combine, split and simplify expressions.
- Solve equations with the unknown in the exponent, including ax = b.
No animated video for this topic yet; these notes stand alone.