Logarithms

Hong Kong

Stage 4 - Stage 5

Lesson

Recall that a logarithmic equation of the form $\log_ba=c$`l``o``g``b``a`=`c` is equivalent to an exponential equation of the form $b^c=a$`b``c`=`a`. That is, the value of the logarithm $\log_ba$`l``o``g``b``a` is "the power which $b$`b` must be raised to in order to give $a$`a`".

Using this, if $a$`a` is a rational power of $b$`b` then we can simplify the logarithm nicely. For example,

$\log_28=\log_2\left(2^3\right)=3$`l``o``g`28=`l``o``g`2(23)=3.

In any other case, where $a$`a` is not a rational power of $b$`b`, we can use a calculator (or other technology) to approximate the value. In particular, we can use the logarithm property

$\log_ba=\frac{\log_na}{\log_nb}$`l``o``g``b``a`=`l``o``g``n``a``l``o``g``n``b`

to rewrite any logarithm using a base that is available on the calculator. For example,

$\log_213=\frac{\log_{10}13}{\log_{10}2}\approx3.7004$`l``o``g`213=`l``o``g`1013`l``o``g`102≈3.7004.

Here is a graph of the function $y=\log_2x$`y`=`l``o``g`2`x`:

From the graph, we can see a few features:

- The graph only exists for positive values of $x$
`x`, with an asymptote along the $y$`y`-axis (the line $x=0$`x`=0). - The curve intersects the $x$
`x`-axis at the point $\left(1,0\right)$(1,0). -
The function is increasing, but at a decreasing rate.

In fact, these features are true for any function of the form $y=\log_bx$`y`=`l``o``g``b``x`, where $b>1$`b`>1.

As with other functions that we've looked at, we can transform the graphs of logarithms in a few ways:

- Multiplying by a constant to get $y=a\log_bx$
`y`=`a``l``o``g``b``x`corresponds to vertically scaling the graph by a factor of $a$`a`. This can also involve a reflection about the $x$`x`-axis if $a$`a`is negative. For example, here is a graph of $y=-5\log_2x$`y`=−5`l``o``g`2`x`:

- Adding a constant to get $y=\log_bx+k$
`y`=`l``o``g``b``x`+`k`corresponds to vertically translating the graph by $k$`k`units. This translation is up if $k$`k`is positive and down if $k$`k`is negative. For example, here is a graph of $y=\log_2x+5$`y`=`l``o``g`2`x`+5:

- We can also obtain the graph of $y=\log_b\left(-x\right)$
`y`=`l``o``g``b`(−`x`) which corresponds to a reflection about the $y$`y`-axis. For example, here is a graph of $y=\log_2\left(-x\right)$`y`=`l``o``g`2(−`x`):

Note that in all of these cases, the asymptote of the function is still along the $y$`y`-axis. Additionally, a function of the form $y=a\log_bx$`y`=`a``l``o``g``b``x` still intercepts the $x$`x`-axis at the point $\left(1,0\right)$(1,0).

Graphs of the form $y=a\log_bx+k$`y`=`a``l``o``g``b``x`+`k`

To sketch the graph of a function of the form $y=a\log_bx+k$`y`=`a``l``o``g``b``x`+`k`, start by identifying the locations of the asymptote and the $x$`x`-intercept.

Then plot a couple of extra points to see the general shape of the graph.

Consider the function $y=3\log_2x$`y`=3`l``o``g`2`x`.

Solve for the $x$

`x`-coordinate of the $x$`x`-intercept.Complete the table of values for $y=3\log_2x$

`y`=3`l``o``g`2`x`.$x$ `x`$\frac{1}{2}$12 $1$1 $2$2 $4$4 $y$ `y`$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ State the equation of the vertical asymptote.

Sketch the graph of $y=3\log_2x$

`y`=3`l``o``g`2`x`.Loading Graph...

Consider the function $y=\log_3x-1$`y`=`l``o``g`3`x`−1.

Solve for the $x$

`x`-coordinate of the $x$`x`-intercept.Complete the table of values for $y=\log_3x-1$

`y`=`l``o``g`3`x`−1.$x$ `x`$\frac{1}{3}$13 $1$1 $3$3 $9$9 $y$ `y`$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ State the equation of the vertical asymptote.

Sketch the graph of $y=\log_3x-1$

`y`=`l``o``g`3`x`−1.Loading Graph...

Consider the function $y=3\log_4\left(-x\right)$`y`=3`l``o``g`4(−`x`).

Solve for the $x$

`x`-coordinate of the $x$`x`-intercept.Complete the table of values for $y=3\log_4\left(-x\right)$

`y`=3`l``o``g`4(−`x`).$x$ `x`$-16$−16 $-4$−4 $-1$−1 $-\frac{1}{4}$−14 $y$ `y`$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ State the equation of the vertical asymptote.

Sketch the graph of $y=3\log_4\left(-x\right)$

`y`=3`l``o``g`4(−`x`).Loading Graph...