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 function g is defined by \( g\left(x\right)=10^{4x-2} \) . What is the value of x when g(x) is equal to 10,000?
原库参考解析
指数式・指数方程与函数特征挑战
For the exponential function f , the value of f(1) is k, where k is a constant. Which of the following equivalent forms of the function f shows the value of k as the coefficient or the base?
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(3.1x+3.1b), 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(3.1x＋k)
II. h(x)= a(3.1)x + m
原库参考解析
指数式・指数方程与函数特征进阶
The table shows three values of x and their corresponding values of y for the equation \( y=7(2)^{x}+3. \)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 function f is defined by \( f\left(x\right)=4^{\frac{x}{6}}+12 \) . For what value of x is f (x) equal to 20?
原库参考解析
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 \(x\ge0\). Which of the following equations displays, as a constant or coefficient, the maximum value of the function it defines, where \(x\ge0\)?
I. \(f(x)=17(1.26)^x+41\)
II. \(g(x)=6(0.78)^x\)
原库参考解析
指数式・指数方程与函数特征挑战
\( f\left(x\right)=336\left(2\right)^x \)
The function f is defined by the given equation. If \( g\left(x\right)=f\left(x-4\right) \) , which of the following equations defines the function g?
原库参考解析
指数式・指数方程与函数特征进阶
\( f\left(x\right)=8\left(2\right)^{\frac{x}{4}} \)
Which table gives four values of x and their corresponding values of \( f\left(x\right) \) for the given exponential function?
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\( f(x)=ab^{\frac{x}{n}} \) , where a, b, and n are constants, and b and n are integers, if \( f\left(3\right)=6 \) and \( f\left(5\right)=150 \), what is the value of \( f\left(6\right) \) ?
原库参考解析
指数式・指数方程与函数特征进阶
\( f\left(x\right)=24\left(2\right)^{\frac{x}{6}} \)
Which table gives four values of x and their corresponding values of \( f\left(x\right) \)for the given exponential function?
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 \( f\left(x\right)=ab^{\frac{x}{n}} \) , where a, b, and n are constants, and b and n are integers. If \( f\left(2\right)=6 \) and \( f\left(5\right)=162 \), what is the value of \( f\left(7\right) \) ?
原库参考解析
指数式・指数方程与函数特征进阶
\( f\left(x\right)=-26\left(2\right)^{\frac{x}{6}} \)
Which table gives four values of x and their corresponding values of \( f\left(x\right) \) for the given exponential function?
原库参考解析
