这部分需要会什么？
- 使用指数运算法则改写表达式
- 用同底数解指数方程
- 理解指数函数的初始值与每单位倍率
核心概念与公式
在表达式有定义的前提下，\(a^ma^n=a^{m+n}\)，\(a^m/a^n=a^{m-n}\)，\((a^m)^n=a^{mn}\)。除法和负指数要求底数非零；涉及任意实数指数时通常取正底数。
指数函数 \(y=Ab^x\)（\(A>0,b>0,b\ne1\)）的初始值为 \(A\)，输入每增加 1，输出乘 \(b\)。当 \(b>1\) 时递增；\(0<b<1\) 时递减。指数方程能改成相同正底数且底数不为 1 时，可以比较指数。
做题时怎样使用
- 先检查底数能否写成同一个数的幂
- 区分“幂相乘”与“指数相乘”，逐步简化
- 解释函数时，找出输入增加 1 对输出造成的倍率
容易混淆的地方
\(a^m+a^n\) 一般不能合并成 \(a^{m+n}\)。\(A\) 是乘在指数项外面的系数，不是底数。
用一个例题串起方法
知识点教学例题解方程 \(8^{x-1}=4^{x+1}\)。
展开解题思路 ＋
统一底数为 2：\(2^{3x-3}=2^{2x+2}\)。由于底数为 2，可令指数相等，得 \(3x-3=2x+2\)，所以 \(x=5\)。
本知识点题库
先独立作答，再查看答案与解析。
指数式・指数方程与函数特征进阶
What is the y-coordinate of the y-intercept of the graph of \( y=137(23)^x \) in the xy-plane?
参考答案
指数式・指数方程与函数特征进阶
The function \( f \) is defined by \( f(x)=2(3)^x \). What is the value of \( f(x) \) when \( x=4 \)?
参考答案
指数式・指数方程与函数特征挑战
The function \( f \) is defined by \( f(x)=a^x-b \), where \( a \) and \( b \) are constants. In the xy-plane, the graph of \( y=f(x) \) passes through the points (c, 8) and (2c, 218), where c is a constant. Which of the following could be the value of b?
参考答案
指数式・指数方程与函数特征挑战
The graph of \( y=a^x-b \) is shown, where a and b are constants. What is the value of b?
参考答案
指数式・指数方程与函数特征进阶
For the function h, the table gives three values of x and their corresponding values of h(x). Which equation could define h?
原库参考解析
指数式・指数方程与函数特征进阶
What is the y-intercept of the graph of \( y=12^{x+2} \) in the xy-plane?
原库参考解析
指数式・指数方程与函数特征挑战
The exponential function \( g \) is defined by \( g(x)=5·a^{x} \), where a is a positive constant. If \( g(3)=625 \), what is the value of \( g(5) \)?
原库参考解析
指数式・指数方程与函数特征进阶
For the function \( f \), the table shows four values of x and their corresponding values of \( f(x) \).
Which of the following could describe this function?
原库参考解析
指数式・指数方程与函数特征进阶
The function \( f \) is defined by \( f\left(x\right)=\left(-8\right)\left(2\right)^x+22 \). What is the y-intercept of the graph of \( y=f\left(x\right) \) in the xy-plane?
原库参考解析
指数式・指数方程与函数特征进阶
Which expression is equivalent to \( \left(m^4q^4z^{-1}\right)\left(mq^5z^3\right) \) , where m, q, and z are positive?
原库参考解析
指数式・指数方程与函数特征基础
The exponential function g is defined by \( g\left(x\right)=19\ ·\ a^x \), where a is a positive constant. If \( g\left(3\right)=2,375 \) , what is the value of \( g\left(4\right) \)?
原库参考解析
指数式・指数方程与函数特征进阶
| Time (years) | Total amount(dollars) |
| 0 | 604.00 |
| 1 | 606.42 |
| 2 | 608.84 |
Rosa opened a savings account at a bank. The table shows the exponential relationship between the time t, in years, since Rosa opened the account and the total amount n, in dollars, in the account. If Rosa made no additional deposits or withdrawals, which of the following equations best represents the relationship between t and n?
原库参考解析
