题目

Problem

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

(i) Solve $3^a = 70$ , giving the answer to 3 decimal places.

(ii) Find the exact value of $b$ such that

$4 + 3\log_3 b = \log_3(5b)$

(6)

题目中文翻译

本题必须展示所有解题步骤，不能完全依赖计算器技术。

(i) 解 $3^a = 70$ ，答案保留3位小数。

(ii) 求满足 $4 + 3\log_3 b = \log_3(5b)$ 的 $b$ 的精确值。

（6分）

解答

(i) 对 $3^a = 70$ 两边取对数，利用换底公式求解。 (ii) 利用对数运算法则化简方程，转化为指数方程求解。

(i) Solve $3^a = 70$ :

\log 3^a =&\, \log 70 \\ a \log 3 =&\, \log 70 \\ a =&\, \frac{\log 70}{\log 3} = 3.867 \end{align*}$$ $$\boxed{a = 3.867}$$ **(ii)** Solve $4 + 3\log_3 b = \log_3(5b)$: $$\begin{align*} 4 + \log_3 b^3 =&\, \log_3(5b) \\ 4 =&\, \log_3(5b) - \log_3 b^3 \\ 4 =&\, \log_3 \frac{5b}{b^3} \\ 4 =&\, \log_3 \frac{5}{b^2} \end{align*}$$ Converting to index form: $$\begin{align*} 3^4 =&\, \frac{5}{b^2} \\ 81 =&\, \frac{5}{b^2} \\ b^2 =&\, \frac{5}{81} \\ b =&\, \frac{\sqrt{5}}{9} \end{align*}$$ (Taking the positive value since the argument of a logarithm must be positive.) $$\boxed{b = \frac{\sqrt{5}}{9}}$$