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。
\( g\left(x\right)=\frac{1}{11}\left(6\left(2\right)^x+7\right) \)
The function g is defined by the given equation. For all values of x, the value of \( g(x) \) is greater than k, where k is a constant. What is the greatest possible value of k ?
Which expression represents the product of \( \left(x^{-9}y^6z^5\right) \) and\( \left(x^4z^5+y^8z^{-7}\right) \) ?
What is the y-intercept of the graph of \( y=10^x+22 \) in the xy-plane?
\( f\left(x\right)=4^{-5\left(x+1\right)} \)
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?
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 expression is equivalent to \( x^6(x^3) \) ?
The y-intercept of the graph of \( y=6\left(1.85\right)^x \) in the xy-plane is \( (0,\ y) \). What is the value of y?
The function \( f \) is defined by \( f\left(x\right)=a\left(4\right)^x \), where a is a positive constant. In the xy-plane, the graph of \( y=f\left(x\right) \) passes through the point (0, 13). What is the value of \( f(0) \)?
\( y=9\left(\frac{a}{7}\right)^{x+c}-b \)
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 \( a>7 \) and \( b>c \)?
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, 10) and (2, 13). What is the value of ab?
\( y=3\left(\frac{a}{6}\right)^{x+c}-b \)
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 \( b>c \)?
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. \(f(x)=a(0.63)^{-bx}\)
II. \(g(x)=a(1.37)^{x+2}+b\)
