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挑战
The table shows three values of x and their corresponding values of y for the equation \( y=7(2)^x+8 \). In the table, a is a constant. What is the value of a?
原库参考解析
指数式・指数方程与函数特征进阶
The table below shows three values of x and their corresponding values of y for the equation \( y=6(2)^x+4 \). In the table, a is a constant. What is the value of a?
原库参考解析
指数式・指数方程与函数特征挑战
The function \( f \) is defined by the equation \( f(x)=20^x+b \), where b is a constant. In the xy-plane, the graph of \( y=f(x) \) contains the points \( (0,\ 36) \) and \( (2,\ p+19) \), where p is a constant. What is the value of p?
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 \( y=4^{x+2} \) in the xy-plane?
原库参考解析
指数式・指数方程与函数特征进阶
An exponential function \( f \) is defined by \( f(x)=k(4)^x \), where k is a positive constant. If \( f(-1)=7 \) and \( f(2)=448 \), what is the value of k?
原库参考解析
指数式・指数方程与函数特征基础
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, 11) and (2c, 221), where c is a constant. Which of the following could be the value of b?
原库参考解析
指数式・指数方程与函数特征挑战
The table shows three values of x and their corresponding values of y for the equation \( y=5(2)^x+9 \) . In the table, a is a constant. What is the value of a?
原库参考解析
指数式・指数方程与函数特征进阶
The expression \( \frac{6x^{22}}{18x^5} \) is equivalent to \( \frac{1}{3}x^b \), where b is a constant and \( x>0 \). What is the value of b?
原库参考解析
指数式・指数方程与函数特征进阶
The graph of \( y=f(x)+2 \) is shown, where \( f \) is defined by an equation of the form \( f(x)=ab^x+c \). If a, b, and c are integer constants and \( c\le0 \), which equation could define \( f \)?
原库参考解析
指数式・指数方程与函数特征进阶
The graph of the equation \( y=4^x+k \) is shown, where k is a constant. What is the value of k?
参考答案
指数式・指数方程与函数特征挑战
The functions \( f \) and g are defined by the given equations, where \( x\ge0 \) , a and b are integer constants, \( a<b \) , and \( a>1 \). If \( y=f(x) \) and \( y=g(x) \) are graphed in the xy-plane, which of the following equations displays, as a constant or coefficient, the minimum value of the function it defines, where \( x\ge0 \)?
I. \( f(x)=a(0.54)^{-bx} \)
II. \( g(x)=a(1.46)^{x+2}+b \)
参考答案
