Exponential graphs and logarithmic inverses
| English | 中文 | Pinyin |
|---|---|---|
| logarithm/ˈlɒɡərɪθəm/ | 对数 | duì shù |
An exponential maps an exponent to a positive amount. Which graph reverses that input-output pair?
- An exponential maps an exponent to a positive amount. Which graph reverses that input-output pair?
- This lesson studies logarithm 对数: The exponent to which a stated valid base must be raised to produce a positive input.
Choose the mathematical structure
- 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.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines logarithm?
The exponent to which a stated valid base must be raised to produce a positive input.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
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).
2^(−3)=1/8, so the logarithm is −3.
Test a tempting shortcut
- 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.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The constant exponential 1^x has a logarithmic inverse for all positive inputs. This claim is false. Explain which definition or assumption it violates.
Find ln(e³).
ln is the inverse of e raised to a power, so ln(e³)=3.
The constant exponential 1^x has a logarithmic inverse for all positive inputs.
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.
Interpret a new situation
- 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.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find e^(ln5).
The positive input 5 lies in the domain of ln, so e^(ln5)=5.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · F. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The exponent to which a stated valid base must be raised to produce a positive input. Choose the relationship, show the method, check its assumptions and interpret the result.