Logarithmic Functions

Lesson

The function that assigns to $y$`y` the logarithm of $x$`x` is called the logarithm function. It is described by the equation $y=\log_b\left(x\right)$`y`=`l``o``g``b`(`x`) where the argument $x$`x` is restricted to the domain of positive reals and the base $b$`b` is any positive real number not equal to $1$1.

If the base $b$`b` is greater than $1$1 then the function monotonically increases across the entire domain $x>0$`x`>0. For $00<`b`<1 the function monotonically decreases across its domain.

Because the logarithm of $1$1 is $0$0, irrespective of the base used, then the curve of all log functions of the form $y=\log_b\left(x\right)$`y`=`l``o``g``b`(`x`) intersects the $x$`x` axis at $(1,0)$(1,0).

The following graph shows $y=\log_2\left(x\right)$`y`=`l``o``g`2(`x`) and $y=\log_{\frac{1}{2}}\left(x\right)$`y`=`l``o``g`12(`x`) illustrating the distinctive shape of the log curve.

Two particular log curves from the family of log functions with $b>1$`b`>1 are shown below. The red curve is the curve of the function $f(x)=\log_2\left(x\right)$`f`(`x`)=`l``o``g`2(`x`), and the blue curve is the curve of the function $g(x)=\log_4\left(x\right)$`g`(`x`)=`l``o``g`4(`x`).

The two points shown on each curve shown help explain the way the gradient of the curve changes as $b$`b` increases in value.

At $x=16$`x`=16 we have $f(16)=\log_2\left(16\right)=4$`f`(16)=`l``o``g`2(16)=4 because $2^4=16$24=16. We also have $g(16)=\log_4\left(16\right)=2$`g`(16)=`l``o``g`4(16)=2 again because $4^2=16$42=16. Using the same reasoning, at $x=\frac{1}{16}$`x`=116 we have $f(\frac{1}{16})=-4$`f`(116)=−4 and $g(\frac{1}{16})=-2$`g`(116)=−2.

More generally we can understand the shape of any log curve by reminding ourselves that if the $x$`x` values grow geometrically in powers (for example $2^0,2^1,2^2,2^3,2^4,2^5...$20,21,22,23,24,25...), then the corresponding $y$`y` values (the logarithms of $x$`x`) grow arithmetically (for example $0,1,2,3,4,5...$0,1,2,3,4,5...).

In a way, we can think of the log function as that function that transforms a geometric world into an arithmetic world - a world of exponents where the vertical $y$`y` numbers are added and subtracted when the horizontal $x$`x` numbers are multiplied and divided. In fact, even though the log function wasn't mathematically defined until much later, this transformation was the key idea that Henry Briggs identified when he invented his base $10$10 logarithms in 1620.

For all allowable values of $b$`b` the function $y=\log_b\left(x\right)$`y`=`l``o``g``b`(`x`) has a vertical asymptote at $x=0$`x`=0. This means that there is no $y$`y` intercept - as $x$`x` approaches $0$0, the function approaches the $y$`y` axis but never arrives there.

The curves of log functions have no horizontal asymptote. If they are growing, they keep growing, but growing at an ever slowing pace. If they are falling, they keep falling, but falling at an ever slowing pace. We could say mathematically that, for $b>1$`b`>1 as $x\rightarrow\infty$`x`→∞, $y\rightarrow\infty$`y`→∞ and for $00<`b`<1 as $x\rightarrow\infty$`x`→∞, $y\rightarrow-\infty$`y`→−∞.

Any table of values normally show the $x$`x` values increasing exponentially as numbers of the form $b^k$`b``k`. For example, to draw $y=\log_{10}\left(x\right)$`y`=`l``o``g`10(`x`) you might construct a table of values with the $x$`x` values as $10^{-3},10^{-2},10^{-1},10^0,10^1,10^2,10^3$10−3,10−2,10−1,100,101,102,103 and the corresponding $y$`y` values as $-3,-2,-1,0,1,2,3$−3,−2,−1,0,1,2,3.

This leads us to discussing some of the ways the logarithm function is used in the modern world. One common application concerns the construction of measurement scales where exponential increases of some physical phenomenon are translated into strictly linear scales.

The Richter scale for measurement of earthquake intensity is a logarithmic scale. An earthquake measuring $6$6 on the Richter scale will have a shaking amplitude (the amplitude of a wave created by a seismograph) $10$10 times that for an earthquake measuring $5$5 on the Richter scale. Further more, the energy ratio between an earthquake of Richter scale $R+1$`R`+1 and $R$`R` is $10\sqrt{10}\approx31.623$10√10≈31.623, so that the energy release of a Richter $6$6 earthquake is over $31$31 times that of a Richter $5$5 earthquake and exactly a thousand times ( $10\sqrt{10}\times10\sqrt{10}$10√10×10√10) a Richter $4$4 earthquake.

As another example, the loudness of a sound, measured in decibels, is a logarithmic scale. For example, suppose a jackhammer has a loudness measure of $120$120 decibels and a normal conversation has a loudness measure of $60$60 decibels. The ratio of the intensity of these two sounds is given by $10^{\frac{120-60}{10}}$10120−6010. In other words a jackhammer is a million times more intense than a normal conversation.

In both the earthquake and loudness measures the log scale becomes a manageable linear scale for exponential phenomena. Of course, you can draw erroneous conclusions when comparing two log scale measurements if you are not aware that they are based on a log scale. For example one could easily think that a jackhammer was twice the intensity of a normal conversation if you were not aware of the specific scaling method used.

Consider the function $y=\log_4x$`y`=`l``o``g`4`x`, the graph of which has been sketched below.

Loading Graph...

Complete the following table of values.

$x$ `x`$\frac{1}{16}$116 $\frac{1}{4}$14 $4$4 $16$16 $256$256 $y$ `y`$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Determine the $x$

`x`-value of the $x$`x`-intercept of $y=\log_4x$`y`=`l``o``g`4`x`.How many $y$

`y`-intercepts does $\log_4x$`l``o``g`4`x`have?Determine the $x$

`x`value for which $\log_4x=1$`l``o``g`4`x`=1.

We are going to sketch the graph of $y=\log_2x$`y`=`l``o``g`2`x`.

Complete the table of values for $y=\log_2x$

`y`=`l``o``g`2`x`.$x$ `x`$\frac{1}{2}$12 $1$1 $2$2 $4$4 $16$16 $y$ `y`$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Plot the first four points found in part (a) on the graph.

Loading Graph...The first point, $\left(\frac{1}{2},-1\right)$(12,−1), has been plotted for you.

Now plot the function $y=\log_2x$

`y`=`l``o``g`2`x`by moving the asymptote and the two other points to appropriate positions.Loading Graph...

Consider the function $y=\log_4x$`y`=`l``o``g`4`x`.

Complete the table of values.

$x$ `x`$\frac{1}{1024}$11024 $\frac{1}{4}$14 $1$1 $4$4 $16$16 $256$256 $y$ `y`$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Consider the behaviour of $\log_4x$

`l``o``g`4`x`as $x$`x`changes.Which of the following statements is correct?

$y=\log_4x$

`y`=`l``o``g`4`x`is a decreasing function.A$y=\log_4x$

`y`=`l``o``g`4`x`is an increasing function.B$y=\log_4x$

`y`=`l``o``g`4`x`increases in some intervals of $x$`x`, and decreases in others.C$y=\log_4x$

`y`=`l``o``g`4`x`is a decreasing function.A$y=\log_4x$

`y`=`l``o``g`4`x`is an increasing function.B$y=\log_4x$

`y`=`l``o``g`4`x`increases in some intervals of $x$`x`, and decreases in others.CWhat value does $\log_4x$

`l``o``g`4`x`approach as $x$`x`approaches $0$0?$-4$−4

A0

B$-\infty$−∞

C$-4$−4

A0

B$-\infty$−∞

CWhat happens when $x=0$

`x`=0?$y=-\infty$

`y`=−∞A$y=-4$

`y`=−4B$y$

`y`is undefinedC$y=-\infty$

`y`=−∞A$y=-4$

`y`=−4B$y$

`y`is undefinedC

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

Apply graphical methods in solving problems