# Questions tagged [exponentiation]

Questions about exponentiation, the operation of raising a base $b$ to an exponent $a$ to give $b^a$.

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### Can decay percent be less than $-100\%$?

I have this problem: $$A=A_0\left(1+\frac{r}{n}\right)^{nt}$$ where $A=100$, $A_0=25$, $n=1$, and $t=2$. This leads to \begin{align*} (1+r)^2=4 &\quad\Rightarrow\quad |1+r|=2\\ &\quad\...
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### Prove $\left( \frac{ n }{ n-1 } \right)^{ n-1 } = \left( \frac{ n-1 }{ n } \right)^{ 1-n }$

I have seen this equation: $$\left( \frac{ n }{ n-1 } \right)^{ n-1 } = \left( \frac{ n-1 }{ n } \right)^{ 1-n }$$ As you can see the numerator switched with the denominator and I wonder how. I know ...
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### Why must the base and argument of a logarithm be positive? [duplicate]

A counter-example (based on my misunderstanding) is, say, if $(-3)^3=-27$ then why is it not the case that $log_{-3}(-27)=3$?
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### Using fast modular exponentiation [closed]

I want to work out the last three digits of 1256^13 using fast modular exponentiation but I'm not clear on how to do this and am very confused about the process. Would really appreciate it if anyone ...
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### Scaling as a repeated shift

I was wondering if the following derivation of the scale operator starting from the shift operator is good. Everybody knows that an operator $T_a$ acting on a function $f$ as $$T_af(x)=f(x+a)$$ can ...
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### Meaning of “$\exp[ \cdot ]$” in mathematical equations [duplicate]

I am reading book "Fuzzy Logic With Engineering Applications, Wiley" written by Timothy J. Ross. I am reading chapter 7 and in this chapter, "Batch Least Squares Algoritm" has been defined. It ...
57 views

### Solve nonlinear equation

How to solve nonlinear equation: $$x+2.1*\frac{100}{1+e^{(10-q(x))/3}}-2=0,\\ here \quad q(x)=\frac{100}{1+e^{(10-x)/3}}$$ Are here any numerical method suitable to solve or any package? I tried ...
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### Sum of digits of sum of digits of powers of 12345

Sum of digits of $12345$ is $1+2+3+4+5=15$. The sum of digits of sum of digits is $1+5=6$. I have plotted the sum of digits of powers of $12345$ with blue dots (x-axis is the power). As the average ...
27 views

### Is there a notation for summing a non-variable set of exponents

Given a static (not variable) range of integers, such as One Through Five, I have this equation. $b^1$ + $b^2$ + $b^3$ + $b^4$ + $b^5$ Is there a shorter way to write that, knowing that this is not ...
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### Is it common to write $f'^{2}(x)$, particularly when $f$ is a trigonometric function?

It has been said many times that the notation for exponentiated or inverted trig functions, e.g.: $$\sin^2(x), \tan^{-1}(x), \csc^3(x)$$ is confusing, ugly, and terrible in general, but nevertheless ...
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### Proving $(ab)^{1/n}=a^{1/n}b^{1/n}$.

In this book: Lang, Serge. "Basic Mathematics" (p. 71), appears this theorem and proof. Theorem 1. Let $a$, $b$ be positive real numbers. Then, $$(ab)^{1/n}=a^{1/n}b^{1/n}$$ Proof. Let $r=a^{1/n}$ ...
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### How do I solve for $t$ in this question? $5^{2t+2}-{100}^{2t}=625$

How do I solve for $t$ in this question? $5^{2t+2}-{100}^{2t}=625$ I have tried to express the LHS in terms of 2 and 5 but I don't seem to find any useful result. Any help will be appreciated. ...
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### Tetration fractal algorithm

In the code example at http://code.activestate.com/recipes/577917-tetration-fractal/ the author seems to implement the operation of a complex number, $z=x+iy$, raised to the power of itself, $z^z$, as ...
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### Simplifying Ramanujan's limited infinite root

Background I was playing around with Ramanujan's infinite root : $$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}}$$ You can write any number as this sequence by limiting the number of square roots in ...
166 views

### On a symmetric equation over the integer lattice that involves the Euler's totient function

I would like to know hints or a proof, or counterexamples, for the conjecture that I've stated in the Question below. I'm interested in this in an attempt to continue the study of a question that I've ...
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### Solving an equation with a variable as both a part of a term and exponent

The problem : $$60 + 10x=2^x$$ I've tried using Logarithms which leads to : $$\log(60+10x)=x\log 2$$ The $\log(60+x)$ is the tricky part