*Stretch / Compression*

*Shifts*

PROPERTY OF LOGS

_{x}(a*b) = log

_{x}(a)+log

_{x}(b)

_{10}(10

^{a}) = a

*horizontal compression*.

*vertical shift*of y = logx. The graph moves up 1.

graph y=logx, y=log(10x), and y=log(100x). how do the graphs compare? use a property of logs to show that the graphs are vertical shifts of one another.

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Eric Y. | SAT PrepSAT Prep

There are a few different ideas you need to know here.

GRAPHING FUNCTION CONCEPTS

When graphing, y = f(x)...

Replacing x with (ax) is a horizontal stretch/compression. y = f(ax)

Multiplying to get y = a*f(x) is a vertical stretch/compression.

You should also remember

For vertical moves, if a>1, the graph is stretching out vertically.

For horizontal moves, if a>1, the graph is compressing in horizontally.

Replacing x with (x-b) is a horizontal shift. Moves entire graph left or right

y = f(x) + b is a vertical shift. Moves entire graph up or down

In this case function y= f(x) is y=logx.

PROPERTY OF LOGS

log_{x}(a*b) = log_{x}(a)+log_{x}(b)

When a base x is not shown, it is commonly assumed base x is 10.

log_{10}(10^{a}) = a

-----------------------------------------

OK so knowing all this, now we can start the problem... Wow.

y = logx AND y = log10x

y = log(10x), we are replacing x with 10x. x>1 Therefore, this is a *horizontal compression*.

y = log10x = log10 + logx

y = 1+logx

y = logx + 1 Therefore, this is *vertical shift* of y = logx. The graph moves up 1.

I know this is a lot of information, but if you take the time to understand all these ideas, you will likely be ahead of your class!

Eric's explanation is confusing because of the typos in it.

So let me try:

f(x) = log(x)

g(x) = log(10x) = log(10^1) + log(x) = log(x) + 1 = f(x) + 1

h(x) = log(100x) = log(10^2) + log(x) = log(x) + 2 = f(x) + 2

So g is f shifted up 1 and h is f shifted up 2.

Eric also explained dilation ("Honey, I shrunk the kids!"), but that's not needed for this problem.

Here’s a GeoGebra graph of the three functions:

http://www.wyzant.com/resources/files/265623/logs_translated

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