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Rates of Growth(a) By drawing the graphs of the functions$$f(x)=1+\ln (1+x) \quad\ and \quad g(x)=\sqrt{x}$$in a suitable viewing rectangle, show that even when alogarithmic function starts out higher than a root function, it is ultimately overtaken by the root function.$$\begin{array}{l}{\text { (b) Find, rounded to two decimal places, the solutions of the }} \\ {\text { equation } \sqrt{x}=1+\ln (1+x)}\end{array}$$

a) See the graph inside.b) $x \approx 13.50$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 3

Logarithmic Functions

Missouri State University

Oregon State University

Harvey Mudd College

Lectures

02:05

Rates of Growth(a) By …

01:23

a. By drawing the graphs o…

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(a) By drawing the graphs …

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Graphical Analysis Use a g…

01:05

Rates of Growth Compare th…

01:55

Use a graphing utility …

04:20

In these exercises we use …

02:46

The ranking of growth rate…

01:37

If the graph of a logarith…

00:14

02:12

Comparing Rates of Growth …

00:47

Logs with different bases …

06:06

(a) Compare the rates of g…

02:57

Sketch the graph of an inc…

09:14

01:24

01:00

Graph each of the followin…

00:55

13:10

The derivative $d t / d x$…

04:47

Compare the functions $f(x…

So here I have graph T two functions affects and genetics. And so our f of X are natural Law function is in red and our g of X function is in is in blue Well, I have graft these two together. And so we noticed that even though the the square root function starts below the the natural log function, we realized that as we approach positive infinity as we approached positivity well f of X is becomes greater then g of X as as X tends to infinity. And so So now what we want to see is where do these two points intersect? In other words, what is this point here? This point here and so So if we have this this if we want to solve for the equation here, if we simplify the equation and we say we have the square, the sward of X minus one and this is the exponents too he is equal to is equal to one plus x. Well, well, now we can say Well, if we have e to the square root of X times one over e minus one minus X. They should give us zero. And so Now, if we saw for this, what we get is that X is X is approximately 13.5. And so this is our solution for when these two graphs intersected. So So let's see if our our answer matches well, we have a graph. Well, if this is, let's say this is 14. Say this is about approximately 14. Well, this is 13.5, and so So this answer does match our graph. And so we are. We are correct.

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Rates of Growth(a) By drawing the graphs of the functions$$f(x)=…

a. By drawing the graphs of the functions $$f(x)=1+\ln (1+x)$$ and $$g(x)=\s…

(a) By drawing the graphs of the functions$$f(x)=1+\ln (1+x) \quad \…

Graphical Analysis Use a graphing utility to graph $f(x)=\ln x$ and $g(x)$ i…

Rates of Growth Compare the rates of growth of the functions $f(x)=\ln x$ an…

Use a graphing utility to graph $ f $ and $ g $ in the same viewing …

In these exercises we use a graphing calculator to compare the rates of grow…

The ranking of growth rates given in the text applies for $x \rightarrow \in…

If the graph of a logarithmic function $f(x)=\log _{a} x,$ where $a>0$ an…

Comparing Rates of Growth Order the functions

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Sketch the graph of an increasing function $f(x)$ such that both $f^{\prime}…

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Graph each of the following functions in the same viewing rectangle and then…

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Height of a Balloon A hot-air balloon is floating above astraight road. …