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
What is an equation of the graph shown?
参考答案
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 \( f\left(x\right) \).
Which of the following could describe this function?
参考答案
指数式・指数方程与函数特征挑战
What is the y-coordinate of the y-intercept of the graph of \( y=(\frac{3}{11})^{x+1} \) in the xy-plane?
参考答案
指数式・指数方程与函数特征挑战
The function h is defined by \( h(x)=a^x+b \), where a and b are positive constants. The graph of \( y=h(x) \) in the xy-plane passes through the points (0, 7) and (2, 22). What is the value of ab?
参考答案
指数式・指数方程与函数特征挑战
The function \( f \) is defined by \( f\left(x\right)\ =\ a\left(2.7^x+2.7^b\right) \) , where a and b are integer constants and \( 0<a<b \) . The functions g and h are equivalent to function \( f \), where k and m are constants. Which of the following equations displays the y-coordinate of the y-intercept of the graph of \( y=f(x) \) in the xy-plane as a constant or coefficient?
I. \( g(x)=a(2.7^{x}+k) \)
II.\( h(x)=a(2.7)^x+m \)
I. \( g(x)=a(2.7^{x}+k) \)
II.\( h(x)=a(2.7)^x+m \)
参考答案
指数式・指数方程与函数特征进阶
What is an equation of the graph shown?
参考答案
指数式・指数方程与函数特征挑战
\( f(x)=4^{-5(x+1)} \)
Which of the following equivalent forms of the given function \( f \) displays, as the base or the coefficient, the y-coordinate of the y-intercept of the graph of \( y=f(x) \) in the xy-plane?
Which of the following equivalent forms of the given function \( f \) displays, as the base or the coefficient, the y-coordinate of the y-intercept of the graph of \( y=f(x) \) in the xy-plane?
参考答案
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?
Which of the following could describe this function?
参考答案
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 \( g(5) \)?
参考答案
指数式・指数方程与函数特征进阶
Which expression is equivalent to \( x^{20}(x^4) \)?
参考答案
指数式・指数方程与函数特征挑战
The function \( f \) is defined by \( f(x)=a(2.1^x+2.1^b) \) , where a and b are integer constants and \( 0<a<b \) . The functions g and h are equivalent to function \( f \), where k and m are constants. Which of the following equations displays the y-coordinate of the y-intercept of the graph of \( y=f(x) \) in the xy-plane as a constant or coefficient?
I. \( g(x)=a(2.1^x+k) \)
II. \( h(x)=a(2.1)^x+m \)
I. \( g(x)=a(2.1^x+k) \)
II. \( h(x)=a(2.1)^x+m \)
参考答案
指数式・指数方程与函数特征进阶
What is the y-intercept of the graph of \( y=16^x \) in the xy-plane?
参考答案
