Logarithmic Functions
Introduced| English | Meaning |
|---|---|
| base/beɪs/ | The number raised to a power in an exponential expression. |
| logarithmic function/ˌlɒɡəˈrɪθmɪk ˈfʌŋkʃn/ | A function giving the exponent to which a fixed positive base other than one must be raised to obtain its positive input. |
| domain/dəˈmeɪn/ | The set of input values for which a function is defined. |
| range/reɪndʒ/ | The set of output values actually attained by a function on its stated domain. |
| vertical asymptote/ˈvɜːtɪkl ˈæsɪmptəʊt/ | A vertical line approached by a curve as its output grows without bound near a particular input. |
| concavity/kənˈkævɪti/ | The direction in which a curve bends, indicated by how its gradient changes. |
| average rate of change/ˈævrɪdʒ reɪt ɒv tʃeɪndʒ/ | The change in a function's output divided by the change in its input across an interval. |
| concave down/kɒnˈkeɪv daʊn/ | Describing a graph whose slopes decrease as the input increases; a negative second derivative is a sufficient condition. |
| concave up/kɒnˈkeɪv ʌp/ | Describing a graph whose slopes increase as the input increases; a positive second derivative is a sufficient condition. |
| end behavior/end bɪˈheɪvjə/ | The way a function behaves as its input becomes very large positively or negatively. |
| point of inflection/pɔɪnt ɒv ɪnˈflekʃn/ | A point on a curve where its concavity changes. |
A display can rise or fall
- Suppose a sensor displays $\log_2 x$ for a positive signal strength $x$. Doubling the signal from 1 to 2 to 4 changes the display from 0 to 1 to 2: equal ratios produce equal output steps.
- A reversed display, $-\log_2 x$, instead shows 0, $-1$, $-2$. A logarithm does not always rise. Its direction depends on both its base 底数 and its scale factor.
State the general form
- A logarithmic function 对数函数 in general form is $f(x)=a\log_b x$, with $a\ne0$, base $b>0$ and $b\ne1$. A zero scale factor would give a constant instead.
- The change-of-base formula gives $f(x)=c\ln x$, where $c=\dfrac{a}{\ln b}$. The sign of $c$ determines whether the graph rises or falls, how it bends and which way its ends go.
Keep the domain separate from the range
- The domain 定义域 is $x>0$, because the logarithm needs a positive argument. The range 值域 is all real numbers when $a\ne0$: outputs may be negative, zero or positive.
- Every unshifted function in this family passes through $(1,0)$ and has no y-intercept. The vertical asymptote 竖直渐近线 is $x=0$, which is outside the domain.
For $b>0$ and $b\ne1$, the domain of $f(x)=\log_b x$ is…
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You can only take the log of a positive number, so the domain is $x > 0$.
For a valid base b, the unshifted function $f(x)=\log_b x$ has a ____ asymptote at $x=0$.
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The function is unbounded near x=0 from the positive side. The limit is −∞ for b>1 and +∞ for 0<b<1, so x=0 is a vertical asymptote in both cases.
For a valid base b, the graph of $y=\log_b x$ crosses the x-axis at…
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Since $b^0 = 1$, we have $\log_b 1 = 0$: the x-intercept is always at $x = 1$.
Check the base and scale factor together
- If $c=a/\ln b>0$, the function is increasing. This happens when $a>0,b>1$ or when $a<0,0<b<1$. Thus both $\log_2 x$ and $-\log_{1/2}x$ increase.
- If $c<0$, it is decreasing. This happens when $a<0,b>1$ or when $a>0,0<b<1$. Thus both $-\log_2 x$ and $\log_{1/2}x$ decrease.
Read signs around the intercept
- For an increasing member of this family, outputs are negative on $0<x<1$, zero at 1 and positive on $x>1$. For example, $\log_2(1/4)=-2$ and $\log_2 4=2$.
- For a decreasing member, these signs reverse. In particular, $\log_{1/2}(1/4)=2$ and $\log_{1/2}4=-2$. A negative logarithm output is compatible with a positive input.
Evaluate $\log_{1/2}4$.
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(1/2)⁻²=4, so the output is −2. This decreasing logarithm is negative for x>1.
Use equal input gaps to identify concavity
- Concavity 凹凸性 describes how slopes change. For $\log_2 x$, the average rate of change 平均变化率 over $[1,2]$ is 1; over $[2,3]$ it is about 0.585. These rates illustrate its concave down 凸 shape.
- For $-\log_2 x$, the corresponding rates are $-1$ and about $-0.585$. They are increasing towards zero, so the graph is concave up 凹 even though its values decrease. In general, $c>0$ gives concave down and $c<0$ gives concave up throughout the domain.
Compare direction and concavity with equal input gaps
Start with log₂x over [1,2] and [2,3]. Compare scale factors and bases, then move the vertical asymptote.
How does $f(x)=-\log_{1/2}x$ behave on its domain?
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Both a and ln(b) are negative, so c=a/ln(b)>0. It has the same shape as log₂x.
The function $-\log_2 x$ is concave down because it decreases.
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Direction and concavity describe different things. Its unit-interval rates increase from −1 to about −0.585, so it is concave up.
Follow both ends of the graph
- The end behavior 末端行为 depends on the same sign. For $c>0$, $f(x)\to-\infty$ as $x\to0^+$ and $f(x)\to+\infty$ as $x\to\infty$.
- For $c<0$, the limits reverse: $f(x)\to+\infty$ as $x\to0^+$ and $f(x)\to-\infty$ as $x\to\infty$. Neither case has a horizontal asymptote. A rate approaching zero does not force the outputs to approach a finite limit.
An unshifted nonconstant logarithmic function approaches a finite horizontal asymptote as x grows without bound.
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For c=a/ln(b)>0 the output tends to +∞; for c<0 it tends to −∞. Neither case has a finite horizontal asymptote.
As $x\to0^+$, what happens to $\log_{1/2}x$?
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Here c=1/ln(1/2)<0, so the decreasing logarithm grows without bound as its positive input approaches zero.
Distinguish the full domain from a closed interval
- On $(0,\infty)$, a nonconstant logarithm is strictly increasing or decreasing, so it has no maximum or minimum. A point of inflection 拐点 would require a change of concavity; this family has none.
- On a closed interval within the domain, endpoint extrema do exist. For $f(x)=\log_2 x$ on $[1,8]$, the minimum is $f(1)=0$ and the maximum is $f(8)=3$. For $-\log_2 x$ on the same interval, the maximum is 0 at 1 and the minimum is $-3$ at 8.
Select all true statements about $y = \log_b x$ (with $b > 1$).
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It rises without bound, so it has no maximum. The other three are correct.
What is the maximum value of $f(x)=\log_2 x$ restricted to $[1,8]$?
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The function is increasing, so the maximum is at the right endpoint: f(8)=3.
A function $a\log_b x$ with $a\ne0$, $b>0$ and $b\ne1$ has a point of inflection somewhere in its domain.
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The sign of a/ln(b) is fixed, so the graph keeps one concavity throughout its domain.
Relate proportional inputs to equal output steps
- For $f(x)=a\log_b x$, replacing $x$ by $bx$ changes the output by $a$. With $f(x)=2\log_2 x$, inputs $1,2,4,8$ produce outputs $0,2,4,6$.
- To increase the output by 1, multiply the input by $b^{1/a}$. For $2\log_2 x$, that factor is $\sqrt2$. Equal additive input gaps, used to compare slopes, are a different check from these multiplicative input steps.
Check a shifted argument before using ratios
- For $g(x)=\log_2(x+1)$, the domain is $x>-1$ and the vertical asymptote is $x=-1$. Outputs $1,2,3,4$ occur at inputs $1,3,7,15$, whose successive ratios are not constant.
- The distances from the asymptote, $x+1=2,4,8,16$, are proportional instead. More generally, $F(x)=a\log_b(x-h)$ has domain $x>h$, asymptote $x=h$ and intercept $(h+1,0)$. A nonzero shift preserves direction and concavity, but raw input values no longer have the unshifted proportional pattern.
For an unshifted logarithm, check $a\ne0$, $b>0$ and $b\ne1$. Use the sign of $a/\ln b$ for direction, concavity, signs and end behavior. Keep the domain, any interval restriction and any input shift with the rule.
What is the domain of $g(x)=\log_2(x+1)$?
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The argument must satisfy x+1>0. The vertical asymptote moves to x=−1.
For $g(x)=\log_2(x+1)$, equal output steps imply a constant ratio between the raw input values x.
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Outputs 1,2,3,4 correspond to inputs 1,3,7,15, whose ratios differ. The distances x+1=2,4,8,16 are proportional instead.
Explain two contrasting graphs
- Evaluate $\log_5 1$ and explain the intercept using an exponential statement.
- Compare $\log_5 x$ with $\log_{1/2}x$: state their domain, direction, concavity and behavior near zero and for large positive inputs.
Evaluate $\log_5 1$ using an exponential statement. Compare $\log_5 x$ and $\log_{1/2}x$: give their domain, direction, concavity and behavior near zero and for large positive inputs.
Any allowed base to the power 0 is 1.
What is log base 5 of 1?
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The log of 1 is 0. Other options — 1 (the input); 5 (the base).