Question 172999
they want to know the impact of doubling x.
i take this to mean:
if you have {{{e^x}}} and you double x, then you have {{{e^(2x)}}}
likewise:
if you have {{{log(10,x)}}} and you double x, then you have {{{log(10,2x)}}}
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first i will graph {{{e^x}}} and {{{e^(2x)}}} to see what the impact would be:
look below the graph for further comments.
{{{graph(800,800,-3,5,-2,20,2.718281828^x,2.718281828^(2x))}}}
the faster rising curve is {{{e^(2x)}}}
when x = 0, the values of both curves is the same.
when x = 1, the value of {{{e^x}}} is 2.718... and the value of {{{e^(2x)}}} is 7.389...
when x = 2, the value of {{{e^x}}} is 7.389... and the value of {{{e^(2x)}}} is 54.598...
the value of {{{e^2x}}} appears to be the value of {{{(e^x)^2}}} which is should since{{{(a^b)^c}}} equals {{{a^(b*c)}}} by the laws of exponentiation.
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when x = -.5, {{{e^x}}} = .6065 and {{{e^(2x)}}} = .3678.
{{{.6065^2}}} = .3678 so the ratio holds even though {{{e^(2x)}}} is smaller than {{{e^x}}} at this point.
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you can see this on the graph.
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the graph of {{{log(10,x)}}} and {{{log(10,2x)}}} is shown now.
look below the graph for further comments.
{{{graph(800,800,-2,40,-1,2,log(10,x),log(10,2x))}}}
this graph doesn't show such a dramatic change in the value of log(x) versus log(2x)
example:
when x = 10, log(x) = 1, and log(2x) = 1.301...
when x = 100, log(x) = 2, and log(2x) = 2.301...
when x = 1000, log(x) = 3, and log(2x) = 3.301...
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this is because when x is an exponent, it has a very large impact on the result.
when x is the result of exponentiation, it has a very small impact on the result.
when x is the exponent, this is what happens:
if you take x = 1, then 10^1 = 10
if you take x = 10, then 10^10 = 10000000000
there's a big difference.
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when x is the result of exponentiation, this is what happens:
if you take x = 1000, then 10^y = 1000 which means y = 3.
if you double x to be 2000, then 10^y = 2000 which means y = 3.301029996
doubling the answers causes a very small increase in the exponent.
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y = 10^x means x is the exponent
y = log(x) means x is the answer.
remember:
y = log(x) if and only if x = 10^y
x is the result of exponentiation when you say y = log(x)
x is the exponent itself when you say y = 10^x.
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best i can do.
hope it makes sense.
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