A logarithm is the answer to "what power?" — nothing more.
Evaluate
log₂ 32
log₂ 32 asks "2 to what power gives 32?" Since 2⁵ = 32, the answer is 5. B divides 32 by 2, and C multiplies — both treat the log as an arithmetic operation on the two numbers, which it is not. The log is always the EXPONENT, so it is usually a small number even when the argument is large.
There is no power of 2 that gives 0 or a negative number.
This is where the domain restrictions in the next lesson come from, and why every answer to a logarithmic equation has to be checked.
The laws convert multiplication into addition, not addition into addition. Test it: log₂(4 + 4) = log₂ 8 = 3, while log₂ 4 + log₂ 4 = 2 + 2 = 4. A sum inside a logarithm cannot be broken up at all.
A calculator only has the two named buttons. Everything else comes through the change-of-base formula in the last step.
Questions often give several logs and ask for one. Use the power law first, so every log has a plain coefficient of 1, then combine: plus means multiply, minus means divide.
Write 3 log x − log y as a single logarithm.
3 log x = log(x³)
the 3 climbs up as a power
log(x³) − log y = log(x³y)
minus means divide
Evaluate log₂ 40 − log₂ 5.
= log₂ (405)
combine first
= log₂ 8
neither 40 nor 5 is a power of 2, but 8 is
= 3
2³ = 8
Writing log₂ 40 − log₂ 5 = log₂ 35 subtracts the arguments, and 35 is not even a power of 2. The quotient law divides: 40 ÷ 5 = 8.
A fraction inside: log₃ (19)
19 = 13² = 3−2
one over a power
log₃ (19) = −2
a negative log
Any argument between 0 and 1 gives a negative log when the base is above 1. That is allowed; only the argument must be positive.
A root as the answer: log₄ 2
41/2 = √4 = 2
2 is the square root of 4
log₄ 2 = 12
a fraction log
When the argument is smaller than the base, the log is a fraction. Check: 4 to the half is 2 ✓.
The base is the unknown: logb 125 = 3
b³ = 125
rewrite in exponential form
b = 5
5 × 5 × 5 = 125
Turning the log into its exponential form works whichever of the three numbers is missing.
logb a = log alog b
any common base works
log₃ 50 = log 50log 3
two button presses
= 1.69900.4771 ≈ 3.561
sensible: 3³ = 27, 3⁴ = 81
Always sanity-check a logarithm by bracketing it. Since 50 sits between 3³ and 3⁴, log₃ 50 has to be between 3 and 4. An answer of 1.5 or 17 is wrong before you check the arithmetic.
Periodical exams ask you to convert between log and exponential form, evaluate a log, expand or condense using the laws, and spot the false "law"; entrance tests add fractions and unknown bases, or combine two logs into one whole number. Turn every log into a "what power?" question, and condense before evaluating. For a 10-second check, raise the base to your answer: it must give back the argument.
Evaluate
log₃ 81
The question is "3 to what power gives 81?" Counting up: 3, 9, 27, 81 — that is four steps, so the answer is 4. C is 3³ and D is 3⁵; both are powers of 3, but the log asks for the exponent, not another power.
Hint: write the powers of 3 in a line and count how far along 81 sits. The log is that count.
