Chapter 8

Logarithmic and Exponential Functions (Pure 3)

The laws of logarithms, solving equations where the unknown is an index, the graphs of $e^{x}$ and $ln x$ , and turning experimental data into a straight line to measure its constants.

หลักสูตรแกนกลาง / สสวท.: จำนวนและพีชคณิต (Number and Algebra)ตรงกับ ม.5 ฟังก์ชันเอกซ์โพเนนเชียลและลอการิทึม (เพิ่มเติม)
Common Core (US): F-LE, F-BF
Cambridge: 9709 Pure Mathematics 3 — Logarithmic and exponential functions

8.1 The laws, and solving for an index

The 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

The third is the one that does the work: it brings an unknown index down to ground level where it can be divided out.

ตัวอย่าง

Solve $5^{2 x - 1} = 3^{x + 2}$ , giving your answer to 3 significant figures.

วิธีทำ

1.Take natural logs: $(2 x - 1) ln 5 = (x + 2) ln 3$
2. $2 x ln 5 - ln 5 = x ln 3 + 2 ln 3$
3. $x (2 ln 5 - ln 3) = 2 ln 3 + ln 5$
4. $x = \frac{2 ln 3 + ln 5}{2 ln 5 - ln 3}$

ตอบ $x \approx 1.36$

ข้อควรระวัง

$ln (A + B)$ is not $ln A + ln B$ . The laws turn multiplication into addition, never addition into addition — and applying the log to a sum term by term is the error that makes a whole solution look plausible and be wrong.

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