Exponential graphs and logarithmic inverses
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| logarithm/ˈlɒɡərɪθəm/ | logarit |
Which graph reverses that input-output pair?
- An exponential maps an exponent to a positive amount.
- This lesson studies logarithm 对数: The exponent to which a stated valid base must be raised to produce a positive input.
The method
- For a>0, y=a^x has every real input and positive outputs.
- It passes through (0,1).
- For a>1 it increases; for 0<a<1 it decreases.
- For a=1 it is the constant 1, with no inverse function.
- For a>0 and a≠1, y=log_a x is the inverse of a^x: domain x>0, range all real numbers. ln x is log_e x and reverses e^x.
Why does 1^x have no logarithmic inverse?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
An inverse exists only for a one-to-one function. The constant map collapses every x to 1, so "log base 1" cannot exist; valid bases are a>0, a≠1.
A worked case
The graph 2^x contains (−1,1/2),(0,1),(1,2),(2,4). Its inverse log₂x contains (1/2,−1),(1,0),(2,1),(4,2), reflected in y=x. The graph (1/2)^x gives 2,1,1/2 at x=−1,0,1, while its inverse log_(1/2)x is decreasing. For e^x and ln x, the points (0,1) and (1,0) are reflected partners; ln e=1 and ln(e³)=3. For any real u, ln(e^u)=u; for x>0, e^(ln x)=x. Logarithms approach but do not reach their vertical boundary x=0; e^x approaches y=0 as x tends to negative infinity.
Exponential graphs and logarithmic inverses
For a>0, y=a^x has every real input and positive outputs
State the valid input, plotted variable or time unit before using the logarithmic/exponential relationship.
Find log₂(1/8).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
2^(−3)=1/8, so the logarithm is −3.
Where it goes wrong
- Neither ln0 nor the logarithm of a negative real number exists in the real-number course.
- A negative logarithm output is allowed: ln(1/e)=−1.
- Base 1 is permitted for an exponential constant but not for a logarithm.
- For a base below 1 the exponential approaches zero as x tends to positive infinity, and the inverse has the opposite increasing/decreasing direction from the a>1 case.
Not true: The constant exponential 1^x has a logarithmic inverse for all positive inputs. 1^x is constant (always 1), so it is not one-to-one and has no inverse.
Find ln(e³).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
ln is the inverse of e raised to a power, so ln(e³)=3.
1^x is constant (always 1), so it is not one-to-one and has no inverse.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. 1^x is constant (always 1), so it is not one-to-one and has no inverse.
Applying it
- State the base, domain and range before taking an inverse.
- Swap the coordinates of exact exponential points to construct the log graph.
- Preserve open boundaries and asymptotes.
- The source F3 prints x≥0; the valid logarithm input condition used here is x>0, since a^u is positive and never zero.
Find e^(ln5).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The positive input 5 lies in the domain of ln, so e^(ln5)=5.
Match each pair to the inverse relationship.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The exponential passes through (0,1); its inverse passes through (1,0). Logarithms accept only positive inputs, though their outputs may be negative.
In your course
- AQA A-level Mathematics (7357), specification section F.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Logarithm: The exponent to which a stated valid base must be raised to produce a positive input.
- For a>0, y=a^x has every real input and positive outputs.
- Watch for: neither ln0 nor the logarithm of a negative real number exists in the real-number course.