Inverses of Exponential Functions
| English | Chinese | Pinyin |
|---|---|---|
| inverse | 反函数 | fǎn hán shù |
| logarithmic function | 对数函数 | duì shù hán shù |
| domain | 定义域 | dìng yì yù |
| vertical asymptote | 竖直渐近线 | shù zhí jiàn jìn xiàn |
The exponential's mirror twin
- Every exponential function has a partner that runs it backward.
- Where $2^x$ turns an exponent into a value, its partner turns a value back into the exponent.
- That partner is the logarithm — and geometrically it is a perfect mirror image.
- Understanding the pair together makes both far easier.
Log is the inverse of exponential
- The logarithmic function 对数函数 $\log_b x$ is the inverse 反函数 of the exponential $b^x$.
- They share the same base and undo each other's work.
- $b^x$ takes an exponent to a value; $\log_b x$ takes the value back to the exponent.
- So logs and exponentials are two views of one relationship.
The logarithmic function $\log_b x$ is the ____ of the exponential function $b^x$.
They undo each other, so the log function is the inverse of the exponential with the same base.
Reflection across y = x
- Because they are inverses, their graphs are reflections across the line $y = x$.
- The soaring exponential becomes a slowly-rising logarithm, and vice versa.
- Every point $(a, b)$ on $b^x$ corresponds to $(b, a)$ on $\log_b x$.
- Seeing one graph, you can sketch the other by flipping across $y = x$.
The exponential curve whose mirror is the logarithm
y = 2ˣ
Picture reflecting this curve y = 2ˣ across the line y = x. The mirror image is the graph of y = log₂ x.
Swapped domain, range, and asymptote
- Inverses swap domain and range. The exponential's range $y > 0$ becomes the log's domain 定义域 $x > 0$.
- The exponential's horizontal asymptote $y = 0$ becomes the log's vertical asymptote 竖直渐近线 $x = 0$.
- The log's range is all real numbers — it can be any exponent.
- So the log function exists only for positive inputs, hugging the y-axis on the left.
The exponential has a horizontal asymptote; its inverse, the log, has a ____ asymptote instead.
Reflecting across $y=x$ turns the horizontal asymptote $y=0$ into the vertical asymptote $x=0$.
The domain of $\log_b x$ is $x > 0$, which was the range of the exponential.
Inverses swap domain and range: the exponential's range ($y>0$) becomes the log's domain ($x>0$).
Select all true statements about the exponential and its inverse.
They are inverses, not the same function. The other three correctly describe the relationship.
They undo each other
- Composing a function with its inverse cancels: $\log_b(b^x) = x$ and $b^{\log_b x} = x$.
- So $\log_2(2^7) = 7$ and $10^{\log_{10} 5} = 5$.
- This cancelling is exactly how logs solve equations with the variable in the exponent.
- Apply the log to both sides, and the exponent comes down where you can reach it.
What is $\log_2\!\left(2^5\right)$?
The log undoes the exponential: $\log_2(2^5) = 5$. Inverse functions cancel.
The exponential accepts any real input but only outputs positives; the log is the mirror — it only accepts positive inputs but outputs any real. Do not give a logarithm a zero or negative argument, and do not expect an exponential to ever output one.
Use the inverse relationship to solve $10^x = 1000$.
- Take $\log_{10}$ of both sides: $\log_{10}(10^x) = \log_{10} 1000$.
- The left side cancels to $x$; the right side is $3$ (since $10^3 = 1000$).
- So $x = 3$ — the logarithm undid the exponential.
The logarithmic function is the inverse of the exponential with the same base: their graphs reflect across $y = x$. Domain and range swap, so the log has domain $x > 0$ and a vertical asymptote at $x = 0$. They undo each other, which is how logs solve exponential equations.