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第四章:部分分式 Partial Fraction

第六章:角的形成及单位 Angles and Measurements

第十三章: 方程组 Simultaneous Equations

第十五章:二元一次不等式及线性规划 linear inequality in two variables and linear programming

17.3 对数的换底公式 Change of Base of Logarithms

17.3 Change of Base of Logarithms 对数的换底公式


 

1.

Given log_23 = m, log_52 = n, prove that {log}_6{1}0 = \frac{n+1}{n+mn}.

已知 log_23 = m, log_52 = n, 试证 {log}_6{1}0 = \frac{n+1}{n+mn}.


 

2.

If log_{27}x^2 = a, state log_3x in terms of a.

若log_{27}x^2 = a, 用a 表示 log_3x。

Answer:
1.5a
3.

Given that log_23 = a and log_37 = b. State log_4256 in terms of a and b.

已知log_23 = a and log_37 = b。 以 a 和 b 表示 log_4256。

Answer:
\frac{3+ab}{1+a+ab}
4.

Show that log_2xy = 2log_4x + 2log_4y. Hence or otherwise, find the values of x and y that satisfies the equation log_2xy = 10 and \frac{{log}_4{x}}{{log}_4{y}} = \frac{2}{3} .

试证 log_2xy = 2log_4x + 2log_4y. 据此或其他方法, 求满足方程式 log_2xy = 10 和 \frac{{log}_4{x}}{{log}_4{y}} = \frac{2}{3} 的 x 及 y 之值。

Answer:
16, 64
5.

If x= \log_3{{\frac{16}{9}} , y= \log_{3}\frac{27}{64} and z= \log_{3}\frac{1}{2} , find the values of

若 x= \log_3{{\frac{16}{9}} , y= \log_{3}\frac{27}{64} 和 z= \log_{3}\frac{1}{2},求值:

(a) \frac{1}{2}x + \frac{1}{2}y – z

(b) 2x + y + 2z

Answer:
16, 64
6.

If a=2^{{log}_8{2}7} and b=2^{{log}_4{3}}, find the value of 2a + b^2

若 a=2^{{log}_8{2}7} 及 b=2^{{log}_4{3}} , 求2a + b^2 的值。

Answer:
9

7.

试证 \frac{1}{{log}_{abc}{b}c}\frac{{log}_b{a}\cdot{log}_c{a}}{{log}_b{a}+log{{_c}a}} = 1


 

8.

If \alpha, \beta are the roots of 3(log_2x)2 – 2log_4x + 1 = 0, form a quadratic equation which roots are log_{\alpha}\beta, log_{\beta}\alpha.

\alpha, \beta 是方程式 3(log_2x)2 – 2log_4x + 1 = 0 的根, 试作一个一元二次方程式,使它的两个根是 log_{\alpha}\beta, log_{\beta}\alpha.

Answer:
3x^2 + 5x + 3 = 0