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进阶
Which expression is equivalent to\( \frac{h^{15}q^4}{h^5q^{16}} \)
,where h>0 and q>0
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 \( f\left(x\right)=-39^x \). The function \( g \) is a decreasing linear function. In the \( xy \)-plane, the graphs of \( y=f\left(x\right) \) and\( y=g\left(x\right) \)intersect at two points, (\( h,\ j \)) and (\( k,\ m \)), where \( j \) > \( m. \) When \( g(x)<f(x) \), which of the following must also be true?
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 equations shown, where a and b are integer constants, a < b and b < 0. 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 y-coordinate of the y-intercept of the graph of the corresponding function?
I. \( f\left(x\right)=a\left(4.2\right)^{x+b} \)
II. \( g\left(x\right)=a\left(4.2\right)^x+b \)
原库参考解析
指数式・指数方程与函数特征挑战
What is the y-intercept of the graph of \( y=11^x \) in the xy-plane?
原库参考解析
指数式・指数方程与函数特征进阶
f(t) = 7,000(0.66)t
The given function f models the number of catalogs a company sent to their customers at the end of each year, where t represents the number of years since the end of 1998, and 0< t <5. lf y = f(t) is graphed in the ty-plane, which of the following is the best interpretation of the y-intercept of the graph in this context?
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)= ax - b, where a and b are constants. In the xy-plane, the graph of y = f(x) passes through the points (c, 8) and (2c, 218), 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 function f is defined by \( f\left(x\right)=a^x+b \) , where a and b are constants. In the xy-plane, the graph of y= f(x) has an x-intercept at (2, 0) and a y-intercept at (0, -24). 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 expression \( \frac{x^{20}\left(x-4\right)}{5x^2}+\frac{4x^{20}}{5x^2} \) is equivalent to \( \frac{1}{5}x^c \) , where \( c \) is a constant and \( x \) ＞0. What is the value of \( c \) ?
原库参考解析
指数式・指数方程与函数特征挑战
An acceptable noise criterion rating for the background noise in a laundry room is 50. For a noise criterion rating of 50, the equation y=22(0.997)x-60+47 gives the estimated sound pressure level, y, in decibels, as a function of the octave band center frequency, x, in hertz, where x≥60. Which of the following is the best interpretation of 47 in this context?
原库参考解析
指数式・指数方程与函数特征进阶
The expression \( \frac{x^{20}\left(x-4\right)}{5x^2}+\frac{4x^{20}}{5x^2} \) is equivalent to \( \frac{1}{5}x^c \) , where c is a constant and x > 0. What is the value of c ?
原库参考解析
指数式・指数方程与函数特征挑战
An acceptable noise criterion rating for the background noise in a laundry room is 50. For a noise criterion rating of 50, the equation y = 22(0.997)x-60 + 47 gives the estimated sound pressure level, y, in decibels, as a function of the octave band center frequency, x, in hertz, where x \( \ge \) 60 . Which of the following is the best interpretation of 47 in this context?
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 (2) = 6 and f (4) = 150, what is the value of f (5)?
原库参考解析
