# Find the doubling constant

My book says Find the doubling constant for the exponential function: $$1.05^x$$

I'm not sure how to work this out, and I'm very sorry if this question is stupid, or too easy for this community.

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What is a doubling constant? –  Dennis Gulko Nov 27 '12 at 11:11
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## 1 Answer

Assuming that "doubling constant" means "the value of $x$ for which $1.05^x = 2$" (I have not met the precise terminology before), then you need take logs of both sides:

$$\log(1.05^x) = \log2$$ then $$x\log1.05 = \log2$$ so that $$x = \frac{\log 2}{\log 1.05} = 14.207$$

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I vaguely remember having seen that terminology before (in I think some dynamical systems or physical context). It is sort of the opposite of half-life. So I think your interpretation is sound. –  Willie Wong Nov 27 '12 at 11:13
@Old John: It sounds somewhat more reasonable to assume that it is the value of $x$ for which $1.05^x=2\cdot1.05=2.1$, but I haven't met this terminology before either. –  Dennis Gulko Nov 27 '12 at 11:13
I took it to mean the value of $x$ that makes $1.05^x$ double what it was when $x=0$, but I might be wrong! –  Old John Nov 27 '12 at 11:15
Math expert, i thank you! –  Frederik Witte Nov 27 '12 at 11:16
The "doubling constant" appears in the world of finance in the form of "the rule of 70" which states that if you have, say, an investment with an annual percentage rate of r, then the number of years it will take to double your original investment will be about $70/r$. This magic number comes from the fact that $\ln 2$ is approximately 0.69. –  Rick Decker Nov 27 '12 at 16:34
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