Logarithmic Functions

Lesson

The function defined by $y=\log_b\left[f\left(x\right)\right]+c$`y`=`l``o``g``b`[`f`(`x`)]+`c` has a domain determined by the solution of the inequality given by $f\left(x\right)>0$`f`(`x`)>0. (Remember that the domain is the set of values that $x$`x` can take) The function $f\left(x\right)$`f`(`x`) is said to be the argument of the log function, and it is this that must be kept positive for the log function to be defined.

For example, consider the following two functions and verify their domains from the graph.

The domain of the function $y=\log_2\left(2x-6\right)-1$`y`=`l``o``g`2(2`x`−6)−1 is found by solving the inequality $2x-6>0$2`x`−6>0. Here, $2x>6$2`x`>6, and thus $x>3$`x`>3.

The domain of $y=\ln\left(-x\right)$`y`=`l``n`(−`x`) is found by setting $-x>0$−`x`>0 from which we see that $x<0$`x`<0.

The range (the set of values that the function can take) can be a little more difficult to determine but there are some forms of the log function where the range is trivial. For example, any log function of the form $y=\log_b\left(mx+c\right)+d$`y`=`l``o``g``b`(`m``x`+`c`)+`d` for constants $a,b,c$`a`,`b`,`c` and $d$`d` will have the range given by $y\in\Re$`y`∈ℜ. That is to say there are no restrictions on the range because the the quantity $mx+c$`m``x`+`c` only effects the domain, and the constant $d$`d` translates the curve vertically without effecting the range.

Thus functions like $y=\log_3\left(2x\right)$`y`=`l``o``g`3(2`x`), $y=\log_3\left(1-2x\right)+2$`y`=`l``o``g`3(1−2`x`)+2 or even $y=\log_{\frac{1}{2}}\left(1+x\right)$`y`=`l``o``g`12(1+`x`) all have the range given by $y\in\Re$`y`∈ℜ.

When thinking about the range, it is a good idea to look at the sketch first. In most cases the sketch will alert you to possible changes to the domain and range. You will find that restrictions start to appear when the the arguments become non-linear.

For example the least value of $y$`y` for the function $y=\log_2\left(x^2+2\right)$`y`=`l``o``g`2(`x`2+2) occurs when $x=0$`x`=0 whence $y=1$`y`=1. The function's natural domain includes all real numbers, and the range is given by $y\ge1$`y`≥1 as readily seen by the graph.

Find the domain of $\log\left(3x+9\right)$`l``o``g`(3`x`+9). State your answer as an inequality.

Find the domain of $\log\left(-7x\right)$`l``o``g`(−7`x`). State your answer as an inequality.

Find the domain of $\ln\left(x^2+9\right)$`l``n`(`x`2+9). Write your answer in interval notation.

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