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- Identify the base.
- Determine points on the graph.
- Plot the points and sketch the graph.

Let's do it!

Looking at the given functions, we can see that we have a common logarithm and natural logarithm. Recall that the base of the common logarithm is $10,$ and the natural logarithm has the number $e$ as its base. With this in mind, let's highlight the base of each function. $f(x)=g_{2}xg(x)=lnxh(x)=gx ⇔⇔⇔ f(x)=g_{2}xg(x)=g_{e}xh(x)=g_{10}x $

Using the base $b$ we can identify three points on the graph of a logarithmic function. $(b1 ,-1),(1,0),and(b,1) $ Since we know the bases, we can immediately determine these points.

Function | Points |
---|---|

$f(x)=g_{2}x$ | $(21 ,-1),$ $(1,0),$ and $(2,1)$ |

$g(x)=g_{e}x$ | $(e1 ,-1),$ $(1,0),$ and $(e,1)$ |

$h(x)=g_{10}x$ | $(101 ,-1),$ $(1,0),$ and $(10,1)$ |

Finally, we will plot the points and connect them with a smooth curve.

b

We can see that, for the three functions, the $x-$variable takes values *greater than* $0,$ and the $y-$values are all real numbers. Therefore, the three functions have the same domain and range.
$Domain:Range: x>0All real numbers $
We can also see that, for the three functions, their values approach infinity as $x$ approaches infinity, and their values approach negative infinity as $x$ approaches $0$ on the positive side. The line $x=0$ is the vertical asymptote of the three graphs. None of them have a horizontal asymptote.
$Verticalasymptotex=0 Horizontalasymptotenone $
The $x-$intercept is the same for the three graphs, and it occurs at $(1,0).$ None of the graphs intercept the $y-$axis, as it is a vertical asymtpote. The three graphs increase in their entire domain, and are negative for values of $x$ *greater than* $0$ **and** *less than* $1.$ Lastly, the three graphs are positive for $x-$values *greater than* $1.$
$x-intercepts:y-intercepts:Increasing Interval:Decreasing Interval:Positve Set:Negative Set: 1nonex>0∅x>10<x<1 $