【实践与探究】
【类比学习】在一次数学兴趣小组活动中,老师和几个同学一起探讨:在
中,
三者的关系.
同学甲:在
中,已知
,求
,这是我们学过的乘方运算,其中
叫做
的
次方.
如:
,则
是
的3次方.
同学乙:在
中,已知
,求
,这是我们学过的开方运算,其中
叫做
的
次方根,
如:
,则
是4的二次方根(即平方根);
,则
是
的三次方根(即立方根).
老师:两位同学说的很好,那么请大家类比平方根、立方根的定义计算:
(1)81的四次方根等于______,
的五次方根等于______;
同学丙:老师,在
中,如果已知
和
,那么如何求
呢?又是一种什么运算呢?
老师:这个问题问的好,已知
,可以求
,它是一种新的运算,称为对数运算.
这种运算的定义是:若
,则
叫做以
为底
的对数,记作:
.
例如:
,则3叫做以2为底8的对数,记作
.
结合上面的学习,请你计算:
(2)
______,
______;
随后,老师和同学们又一起探究出对数运算的一条性质:如果
,
,
,
,那么
.
(3)请利用上述性质计算:
.
【类比学习】在一次数学兴趣小组活动中,老师和几个同学一起探讨:在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ce1edece9463e1958e5e3fed42948a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5d36945c17990eddc1a7247fe9bff6e.png)
同学甲:在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ce1edece9463e1958e5e3fed42948a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/378c3e88252d3daed58e608e82be30de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
如:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08846208b092ed3eb421e3e6ed7daad4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6b9a5f1362c12e1ea8f6fc0d9d787ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274a9dc37509f01c2606fb3086a46f4f.png)
同学乙:在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ce1edece9463e1958e5e3fed42948a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6a6fb7b84bf5f1e0ee82d0a054603b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
如:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30a0fc94bda3fc1c0bd511846050ae5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72b8bee40319fe80e512d221cfe252a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08846208b092ed3eb421e3e6ed7daad4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274a9dc37509f01c2606fb3086a46f4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6b9a5f1362c12e1ea8f6fc0d9d787ac.png)
老师:两位同学说的很好,那么请大家类比平方根、立方根的定义计算:
(1)81的四次方根等于______,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c1dfc770fc27e0b9bf602ec2f842ae9.png)
同学丙:老师,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ce1edece9463e1958e5e3fed42948a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
老师:这个问题问的好,已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/663a61ad241d5d874c9a9362f0ee917c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
这种运算的定义是:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/827a8a66a4bd38f71300d5dd55fa0b7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/267e53555d33f8e6d39f5bafc610f304.png)
例如:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/feae878a430a1b23774127547d842cb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/543fa7a3983d368c56acf31b2d454849.png)
结合上面的学习,请你计算:
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b2581085686d23ac7e7ef64fc1f9a62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86108ccf3a39fefced28e4cfe4373222.png)
随后,老师和同学们又一起探究出对数运算的一条性质:如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c400a615a16a1662de98dfb4e49d58d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2480f87a11c4cd450bc9454ea7276722.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2a891d21bb2c7a11304beaab5054074.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86f74b66b4fc6d478eb4472ee863e09e.png)
(3)请利用上述性质计算:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24704ceed0ec223b6157fd54f2080cd9.png)
更新时间:2024-05-26 14:44:52
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