Logarithmic and Exponential Functions

Logarithms as the inverse of a power, the three laws that do all the work, and the linear law — bending a curved relationship into a straight line so its constants can be measured.

หลักสูตรแกนกลาง / สสวท.: จำนวนและพีชคณิต (Number and Algebra)ตรงกับ ม.5 ฟังก์ชันเอกซ์โพเนนเชียลและลอการิทึม (เพิ่มเติม) — ส่วน "กฎเชิงเส้น" ไม่มีในหลักสูตรไทย
Common Core (US): F-LE, F-BF
Cambridge: IGCSE 0606 — Topic 6: Logarithmic and exponential functions

4.1 A logarithm is an index

The definition everything else follows from

y = a^{x} ⟺ x = lo g_{a} y a > 0, a \neq = 1

Read $lo g_{a} y$ as "the power you raise $a$ to in order to get $y$ ". Every log question gets easier the moment it is read that way.

The three laws

lo g_{a} (M N) = lo g_{a} M + lo g_{a} N lo g_{a} (\frac{M}{N}) = lo g_{a} M - lo g_{a} N lo g_{a} (M^{k}) = k lo g_{a} M

They are the index laws seen from the other side: multiplying numbers adds their indices, so multiplying numbers adds their logs.

Change of base, and the natural logarithm

lo g_{a} b = \frac{lo g _{c} b}{lo g _{c} a} ln x = lo g_{e} x, e \approx 2.71828

ตัวอย่าง 1

Solve $lo g_{2} x + lo g_{2} (x - 2) = 3$ .

วิธีทำ

1.Combine: $lo g_{2} (x (x - 2)) = 3$
2.Undo the log: $x^{2} - 2 x = 8$
3. $x^{2} - 2 x - 8 = 0$ , so $(x - 4) (x + 2) = 0$
4. $x = - 2$ would make $lo g_{2} (- 2)$ undefined — reject it

ตอบ $x = 4$

ตัวอย่าง 2

Solve $3^{2 x + 1} = 20$ , giving your answer to 3 significant figures.

วิธีทำ

1.Take logs of both sides: $(2 x + 1) ln 3 = ln 20$
2. $2 x + 1 = \frac{ln 20}{ln 3} \approx 2.7268$
3. $2 x \approx 1.7268$

ตอบ $x \approx 0.863$

ข้อควรระวัง

Rejecting a root earns a mark; it is not an afterthought. The domain of $lo g_{a} x$ is $x > 0$ , so any candidate that puts zero or a negative inside a log has to be discarded and SAID to be discarded.

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