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。
Which expression is equivalent to \( \left(5x^6\right)\left(x^3\right) \) ?
A beaker containing a liquid is placed on a table. The function \( g\left(t\right)=295+\left(361-295\right)\left(2.72\right)^{-0.104t} \) gives the approximate temperature, in kelvins, of the liquid t minutes after the beaker was placed on the table. According to this function, what was the approximate temperature, in kelvins, of the liquid when the beaker was placed on the table?
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?
\( y=9\left(\frac{a}{6}\right)^{x+c}-b \)
How many times does the graph of the given equation in the xy-plane cross the x-axis, where a, b, and c are positive constants such that \( a>6 \) and\( b>c \) ?
An exponential function f is defined by \( f\left(x\right)=c^x \), where c is a constant greater than 1. If \( f(6)=9·f(4) \), what is the value of c ?
The exponential function f is defined by\( f\left(x\right)=ab^x \), where a and b are positive constants. If \( f\left(n-1\right)=f\left(n\right)+\frac{89}{100}f\left(n-1\right) \), where n is a constant, what is the value of b?
Which expression is equivalent to \( \frac{6v^{10}}{24v^5} \) , where \( v>0 \) ?
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. \(f(x)=19(1.27)^x+43\)
II. \(g(x)=8(0.77)^x\)
The y-intercept of the graph of \( y=81.65^x \) in the xy-plane is (0, y). What is the value of y?
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. \(f(x)=16(1.24)^x+43\)
II. \(g(x)=8(0.72)^x\)
The function \( f \) is defined by \( f\left(x\right)=a^x+b \), where a and b are constants and a > 0. In the xy-plane, the graph of \( y=f\left(x\right) \) has a y-intercept at \( \left(0,-28\right) \) and passes through the point (2, 35). What is the value of a + b?
The functions f and g are defined by the given equations below, where x ≥ 0. Which of the following equations displays, as a constant or coefficient, the minimum value of the function it defines, where x ≥ 0?
Ⅰ.\( f\left(x\right)=14\left(1.4\right)^{x+5} \)
Ⅱ.\( g\left(x\right)=14\left(1.16\right)\left(1.4\right)^{x+3} \)
