Question

若 $\log_{10}2=a$ 且 $\log_{10}3=b$ ，则 $\log_5 12$ 等于：

If $\log_{10}2=a$ and $\log_{10}3=b,$ then $\log_5 12$ equals:

$\dfrac{a+b}{a+1}$

$\dfrac{2a+b}{a+1}$

$\dfrac{a+2b}{1+a}$

$\dfrac{2a+b}{1-a}$

$\dfrac{a+2b}{1-a}$

对常用对数使用换底公式

Use the change-of-base formula with common logarithms

写出 $12=2^2\cdot3$ 和 $5=\frac{10}{2}$

Write $12=2^2\cdot3$ and $5=\frac{10}{2}$

由换底公式， $\begin{aligned} \log_5 12 &=\frac{\log12}{\log5}\\ &=\frac{2\log2+\log3} {\log10-\log2}\\ &=\frac{2a+b}{1-a} \end{aligned}\text{。}$

因此，正确答案是 D。

By change of base, $\begin{aligned} \log_5 12 &=\frac{\log12}{\log5}\\ &=\frac{2\log2+\log3} {\log10-\log2}\\ &=\frac{2a+b}{1-a}. \end{aligned}$

Therefore, the correct answer is D.