Why are logarithmic scales useful?

They let you display data spanning a very wide range of values on a single readable scale. Earthquake magnitudes, sound intensity in decibels and the pH scale in chemistry all work this way.

Why does a logarithmic function have no extrema and no points of inflection?

Because it is always increasing or always decreasing, so it never turns around. Its graph is also always concave up or always concave down, so the concavity never changes and no point of inflection can occur.

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Logarithmic Functions (College Board AP® Precalculus): Flashcards

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Define logarithm.

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Cards in this collection (40)

Define logarithm.
A logarithm answers the question: what power must a base be raised to in order to give a particular value?
In $\log_{b} c$ , the value of the expression is the exponent that $b$ must be raised to in order to produce $c$ .
The statement $\log_{b} c = a$ can be rewritten in exponential form. Fill in the two gaps, the first of which is an exponent.
$b^{______} =______$
The completed exponential form is:
$b^{a} = c$
Logarithmic form and exponential form are two equivalent ways of writing the same relationship, so you can move between them whenever one is more convenient.
What restrictions apply to $b$ and $c$ in $\log_{b} c$ ?
The base must satisfy $b > 0$ and $b \neq 1$ .
The value $c$ must be positive, since you cannot take the logarithm of zero or of a negative number.
What does $\log 100$ mean when no base is written?
It means the common logarithm, which has base $10$ , so $\log 100$ is $\log_{10} 100$ .
Its value is $2$ , because $10^{2} = 100$ .
Evaluate $\log_{4} \frac{1}{16}$ and $\log_{8} 2$ without a calculator.
$\log_{4} \frac{1}{16} = - 2$ and $\log_{8} 2 = \frac{1}{3}$ .
The first holds because $4^{- 2} = \frac{1}{16}$ , and the second because $8^{\frac{1}{3}} = \sqrt[3]{8} = 2$ .
A logarithm is negative when the value is a reciprocal power of the base, and fractional when it is a root of the base.
True or False?
Every logarithm can be evaluated exactly by hand.
False.
Some can, such as $\log_{2} 32 = 5$ , because $2^{5} = 32$ .
Others have no exact value you can reach by hand and must be estimated with technology, such as $\log_{2} 5 = 2.321928 \dots$
Define logarithmic scale.
On a logarithmic scale, each unit along the scale represents a multiplicative change by the base of the logarithm.
On a base- $10$ scale the marks sit at $10^{0}, 10^{1}, 10^{2}, 10^{3}, \dots$ , so each step multiplies the value by $10$ .
On a logarithmic scale, what does equal spacing between two pairs of values tell you?
That the two pairs have equal ratios, rather than equal differences.
The distance from $10$ to $100$ is the same as the distance from $100$ to $1000$ , because each represents multiplying by $10$ .
Why are logarithmic scales useful?
They let you display data spanning a very wide range of values on a single readable scale.
Earthquake magnitudes, sound intensity in decibels and the pH scale in chemistry all work this way.
Fill in the two gaps.
A logarithmic function has general form $f (x) = a \log_{b} x$ , in which $b$ is the $______$ with $b > 0$ and $b \neq 1$ , and $a$ is a nonzero constant that scales the function $______$ by a factor of $a$ .
The completed sentence is:
A logarithmic function has general form $f (x) = a \log_{b} x$ , in which $b$ is the base with $b > 0$ and $b \neq 1$ , and $a$ is a nonzero constant that scales the function vertically by a factor of $a$ .
When $a = 1$ this reduces to the basic logarithmic function $f (x) = \log_{b} x$ .
What is the difference between the expression $\log_{b} c$ and the function $f (x) = \log_{b} x$ ?
The expression $\log_{b} c$ is a single number, the exponent that produces $c$ from the base $b$ .
The function lets the input vary, giving an output for every allowed value of $x$ .
How does the size of the base affect how quickly $\log_{b} x$ grows?
A larger base makes the function grow more slowly, because a larger base needs a smaller exponent to reach the same value.
For example $\log_{10} 100 = 2$ , while $\log_{2} 100 \approx 6.644$ .
Which point does every function $f (x) = \log_{b} x$ pass through?
Every one passes through $(1, 0)$ .
That is because $\log_{b} 1 = 0$ for any allowed base, since $b^{0} = 1$ .
Define the natural logarithm.
The natural logarithm, written $\ln x$ , is the logarithm with the natural base $e$ .
So $\ln x$ is simply shorthand for $\log_{e} x$ .
Without a calculator, find $\ln e$ , $\ln 1$ and $\ln (e^{3})$ .
$\ln e = 1$ , $\ln 1 = 0$ and $\ln (e^{3}) = 3$ .
Each follows from asking what power of $e$ gives the value, since $e^{1} = e$ , $e^{0} = 1$ and $e^{3} = e^{3}$ .
True or False?
The natural logarithm follows different rules from logarithms with other bases.
False.
The natural logarithm follows all the same rules as any other logarithm.
The only difference is that its base is $e$ rather than some other number.
How are $f (x) = \log_{b} x$ and $g (x) = b^{x}$ related?
They are inverse functions of each other, for any base with $b > 0$ and $b \neq 1$ .
Composing them in either order returns the input, so $f (g (x)) = g (f (x)) = x$ .
Fill in the two gaps.
Taking the logarithm base $b$ of $b^{x}$ returns the $______$ itself, and raising $b$ to the power $\log_{b} x$ returns the $______$ value inside the logarithm.
The completed sentence is:
Taking the logarithm base $b$ of $b^{x}$ returns the exponent itself, and raising $b$ to the power $\log_{b} x$ returns the original value inside the logarithm.
In symbols, $\log_{b} (b^{x}) = x$ and $b^{\log_{b} x} = x$ .
An exponential function turns equal additive input changes into equal multiplicative output changes. What does a logarithmic function do?
The reverse: it turns equal multiplicative input changes into equal additive output changes.
Doubling $x$ in $\log_{2} x$ always adds $1$ to the output, just as adding $1$ to $x$ in $2^{x}$ always doubles it.
True or False?
Since $2^{3} = 8$ , the point $(8, 3)$ lies on the graph of $\log_{2} x$ .
True.
Because $2^{3} = 8$ , the point $(3, 8)$ lies on the graph of $2^{x}$ , and the inverse exchanges the two coordinates.
In general, if $(s, t)$ is on the graph of $b^{x}$ then $(t, s)$ is on the graph of $\log_{b} x$ .
The graph of $\log_{b} x$ is the reflection of $b^{x}$ over $y = x$ . What does that do to their asymptotes?
The exponential function's horizontal asymptote at $y = 0$ becomes the logarithmic function's vertical asymptote at $x = 0$ .
Reflecting over $y = x$ exchanges the roles of the two axes, so it exchanges the two kinds of asymptote as well.
How can you rewrite $b^{x}$ as a power of a different base $c$ ?
Use the identity $b^{x} = c^{(\log_{c} b) (x)}$ .
It works because $b$ itself can be written as $c^{\log_{c} b}$ , and raising that to the power $x$ multiplies the exponents.
Rewrite $7^{x}$ in the form $e^{k x}$ , giving $k$ to three decimal places.
$7^{x} = e^{1.946 x}$ .
The identity $b^{x} = e^{(\ln b) (x)}$ gives $7^{x} = e^{(\ln 7) (x)}$ , and $\ln 7 = 1.945910 \dots$
In the identity $b^{x} = c^{(\log_{c} b) (x)}$ , what does $(\log_{c} b) (x)$ mean?
It means the number $\log_{c} b$ multiplied by the variable $x$ .
It does not mean $\log_{c} (b x)$ , which would be the logarithm of the product $b x$ .
Fill in the two gaps.
For $f (x) = a \log_{b} x$ , the domain is all $______$ real numbers and the range is $______$ real numbers.
The completed sentence is:
For $f (x) = a \log_{b} x$ , the domain is all positive real numbers and the range is all real numbers.
The restriction on the domain exists because $\log_{b} x$ is not defined when $x \leq 0$ .
For $f (x) = a \log_{b} x$ , when is the function increasing and when is it decreasing?
With $a > 0$ it is increasing when $b > 1$ , and decreasing when $0 < b < 1$ .
With $a < 0$ both of those are reversed, so a negative coefficient flips the direction.
Why does a logarithmic function have no extrema and no points of inflection?
Because it is always increasing or always decreasing, so it never turns around.
Its graph is also always concave up or always concave down, so the concavity never changes and no point of inflection can occur.
What happens to $\log_{b} x$ with $b > 1$ as $x$ approaches $0$ , and how do you write it?
The output decreases without bound, so there is a vertical asymptote at $x = 0$ , written as:
$\lim_{x \to 0^{+}} \log_{b} x = - \infty$
The $0^{+}$ means approaching from the right only, which is necessary because the function is not defined for $x \leq 0$ .
True or False?
As $x$ increases without bound, $\log_{b} x$ approaches a finite value.
False.
It increases without bound as well, so $\lim_{x \to \infty} \log_{b} x = \infty$ when $b > 1$ .
It does so very slowly, which is why the graph can look as though it is levelling off when it is not.
The outputs of $g$ rise by $1$ each time while the inputs run $5, 8, 14, 26, 50$ . How can $g$ still be modeled by a logarithmic function?
Subtracting $2$ from every input gives $3, 6, 12, 24, 48$ , whose successive ratios are all $2$ .
A logarithmic function has inputs changing proportionally over equal-length output intervals, so $g$ is a horizontal translation of one, $g (x) = f (x - 2)$ .
Note this runs the opposite way round from the hidden-exponential test, where it is the outputs that turn out to be proportional.
The function is $f (x) = - 3 \log_{5} x$ . Write both of its limits.
The two limits are:
$\lim_{x \to \infty} f (x) = - \infty$
$\lim_{x \to 0^{+}} f (x) = \infty$
Both are the opposite of the usual case, because the negative coefficient reverses the sign of every output.
Fill in the two gaps.
The logarithm of a product is the $______$ of the logarithms, and the logarithm of a quotient is the $______$ of the logarithms.
The completed sentence is:
The logarithm of a product is the sum of the logarithms, and the logarithm of a quotient is the difference of the logarithms.
In symbols, $\log_{b} (x y) = \log_{b} x + \log_{b} y$ and $\log_{b} (\frac{x}{y}) = \log_{b} x - \log_{b} y$ .
What does the power property for logarithms say, and what does it let you do to $\frac{1}{2} \ln x$ ?
It says $\log_{b} (x^{n}) = n \log_{b} x$ , so an exponent inside a logarithm can be brought out as a multiplier in front.
Run in reverse it gives $\frac{1}{2} \ln x = \ln (x^{\frac{1}{2}}) = \ln \sqrt{x}$ .
True or False?
${(\log_{b} x)}^{n} = n \log_{b} x$
False.
The power property applies when the exponent sits on the input, giving $\log_{b} (x^{n}) = n \log_{b} x$ .
Raising the whole logarithm to a power is a different operation, and it does not simplify this way.
Your calculator has only $\ln$ and $\log$ buttons. How can you evaluate $\log_{5} 12$ ?
Use the change of base property, $\log_{b} x = \frac{\log_{a} x}{\log_{a} b}$ .
That gives $\log_{5} 12 = \frac{\ln 12}{\ln 5}$ , and $\frac{\log 12}{\log 5}$ produces the same value.
Why is every horizontal dilation of a logarithmic function equivalent to a vertical translation?
Because the product property splits it: $\log_{b} (k x) = \log_{b} k + \log_{b} x$ .
The term $\log_{b} k$ is a constant, so it shifts the graph of $\log_{b} x$ up or down rather than stretching it sideways.
What does raising the input of a logarithmic function to a power do to its graph?
It produces a vertical dilation, by a factor equal to that power.
The power property gives $\log_{b} (x^{k}) = k \log_{b} x$ , which is the original function multiplied by $k$ .
Why are all logarithmic functions vertical dilations of each other?
By the change of base property, $\log_{b} x = \frac{1}{\log_{a} b} \cdot \log_{a} x$ .
Since $\frac{1}{\log_{a} b}$ is only a constant, changing the base stretches or compresses the graph vertically and never alters its shape.
Write $2 \log_{3} x + \log_{3} 5 - 3 \log_{3} y$ as a single logarithm.
$\log_{3} (\frac{5 x^{2}}{y^{3}})$ .
Use the power property first to move the coefficients inside, giving $\log_{3} (x^{2}) + \log_{3} 5 - \log_{3} (y^{3})$ .
The product property then combines the first two terms and the quotient property brings in the third.
Expand $\log_{10} (\frac{a^{2} c}{b^{3}})$ into separate terms.
$2 \log_{10} a + \log_{10} c - 3 \log_{10} b$ .
The quotient property splits the fraction, the product property splits $a^{2} c$ , and the power property brings each exponent out in front.

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