Questions tagged [exponentiation]

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

Exponentiation is a mathematical operation which produces a power $a^n$ from a base $a$ and an exponent $n$. The objects involved are usually numbers, but the procedure can be generalized to matrices, elements in algebraic structures, sets, etc.

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How to calculate the inverse function of $y = x + \ln(x)$?

I have had incidents in the past where my teacher gives a question on non-calculator practice exams that are impossible to solve without a calculator, where you are stuck in an endless loop of moving logs and e's around with out going…
Snowybluesky
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Reverse Bernoulli's inequality?

For $01$, Bernoulli's inequality asserts that $$(1-x)^r\geq 1-rx.$$ Does the reverse inequality hold if we can put a constant in front of $rx$? E.g., $$(1-x)^r\leq 1-\frac{rx}{2}?$$
pi66
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Can we give an upper bound for the sum over primes $p_{i}$ of $\sin(p_{i} x)$?

Let $x$ be a positive real number. Consider the sum $\sum \sin(p_i x)$ taken over all primes $p_i$ from 2 till $n$. Call this function $f(n,x)$. Can we give good upper and lower bounds of $f(n,x)$ for fixed $n$ or $x$? Is the best known upper bound…
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How to write exponential function for curve that pass 4 or more points

Given 3 points, we could write exponential function for a curve that would pass those points in $y = ax^n + b$ form So I wonder that, could we write a function in exponential form that would pass 4 points, which is not quadratic or cubic bezier Or…
Thaina
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How to find modulus square root?

Given integers $m$, $c$ and $n$. Find $m$ such that $m^2 \equiv c \pmod n $ I used Tonelli-Shanks algorithm to caculate the square root, but in my case $n$ is not a prime number, $n = p^2,\ p$ is a prime number. I read this page. It is said…
mja
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Dealing with non-integral powers on negative numbers

If we are given an expression $(-8)^\frac26$, how do we solve it? If it is $(-8)^\frac13$, we can find the cube root of -8 which is -2. However, if we square it first and find the sixth root, we get +2 (or maybe $\pm2$, which still isn't the same as…
ghosts_in_the_code
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If $x^a$ is $x$ multiplied $a$ times, then how does $x^{-1}$ make sense?

What is the meaning of $x$ raised to any non-positive value? We know that $x^{-a} = \dfrac{1}{x^a}$ and $x^0 = 1$, but where does that come from? What is the proof? Why is this true? What about $x$ raised to a fraction, like say $\frac{1}{3}$? How…
jimpix
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How to evaluate $\log(1 - x)$ in terms of $\log(x)$?

I can do this using the following relation: $$\log(1 - x) = \log(1 - \exp(y))$$ Here $y = \log(x)$ is always a negative number. However, I was wondering whether it's possible to compute $\log(1 - x)$ without using exponentiation.
Aadit M Shah
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Exponentiation with negative base and properties

I was working on some exponentiation, mostly with rational bases and exponents. And I stuck with something looks so simple: $(-2)^{\frac{1}{2}}$ I know this must be $\sqrt{-2}$, therfore must be imaginary number. However, when I applied some…
Harry Hong
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Where does this equation come from?

Since I study 3 years i ask myself very often where does this equation come from? $$e^{i\theta} = \cos(\theta)+i \sin(\theta)$$ Is it found by series expansion?
Rob
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When is $\sqrt{x/y^2}$ equal to $\sqrt{x}/y$?

The solution to the quadratics is given by $r = -\dfrac{b}{2a}\pm\sqrt{\dfrac{b^2-4ac}{4a^2}}$, which is shortened to $r = -\dfrac{b}{2a}\pm\dfrac{\sqrt{b^2-4ac}}{2a}$, but I'm wondering how if this is justified, given that $4a^2$ can be negative if…
Frank Vel
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How does $3n+1$ change the proximity of $n$ to a power of two?

This is part of an attempt to prove Collatz's conjecture. I proved a modification of Collatz's conjecture, where instead of $3n+1$ if $n$ is odd, you do $n+1$. In Collatz's conjecture, if you get to a power of two, you're going to get to one. Let…
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"Exponential Madness" (Gauss's challenge)

From Euler's identity, we see that $e^{i\pi}=-1$ $\Rightarrow e^{2ik\pi}=1$ [squaring both sides]. This equation surely holds for all integers $k$. EDIT: From the second equation we get $e^{1+i2k\pi}=e$ So, $e^{1+i2k\pi}=\left (e^{1+i2k\pi}\right…
StubbornAtom
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Seeming Contradiction With Complex Exponents

I was fooling around while waiting for a page to load and came across the following "contradiction". Let $x=(-1)^{i}$. Then $x^{i}=(-1)^{i\cdot i}=(-1)^{-1}=-1$. Thus, $x=\left(x^{i}\right)^{-i}=(-1)^{-i}=\left((-1)^{i}\right)^{-1}=x^{-1}$.…
Bob Knighton
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Logarithm of >2 numbers

I am learning logarithms and i found that $log(a*b) = log(a)+log(b)$ I tried to apply the same principle for three numbers like $log(a*b*c) = log(a)+log(b)+log(c)$ but it didn't work as i expected. Is there any direct formula to calculate this…
Swatak
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