# Tagged Questions

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### Prove logarithm rules using definition as the inverse exponential

Problem $3$. Show that $\operatorname{exp} : \Bbb R \to (0, \infty)$ is bijective. Its inverse function is called the (natural) logarithm $\log : (0, \infty) \to \Bbb R$. Verify the logarithm ...
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### Proving that the exponential function is bijective

Prove that $\exp: \mathbb{R} \mapsto (0,\infty)$ is a bijection. Okay, so the first part is really easy: injectivity follows directly from writing the exponential function as a series. ...
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### Analytical aptitude - Division of exponentials.

What is the remainder when 6^17 + 117^6 is divided by 7? How to approach these type of questions?
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### Changing an exponential function to logarithmic

I have a question stating that $P=75e^{-0.005t}$ and they want to get t by itself. I used the example $y=2^x = x=log_2(y)$ To find that $-0.005t = 75ln(P)$ So $t=\frac{75ln(P)}{-0.005}$ However ...
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### How to solve $e^{ax}+e^{bx}+e^{cx}+d=0$

How to solve an equation like $e^{ax}+e^{bx}+e^{cx}+d=0$ (i.e. to write $x=...$) where $a,b,c,d$ are fixed non-zero real numbers. I have tried assuming that $x=ln(y)$ for $y>0$ but it goes ...
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### Exponential distribution find parametar

Can someone please help me with this problem? Time of production of one electronic component is given with exponential distribution with parameter λ. If the process lasts less than 3 hours, the ...
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### Find the roots of 2 equations

Show that the equation $e^{-x} = x^2$ has a root between $x=0.70$ and $x=0.71$. I think you have to use natural logs to get rid of the $e$ however after that, i'm not sure how to solve for $x$
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### Finding Taylor approximation for $x^4e^{-x^3}$

I'm trying to find Taylor approximation for the function: $$x^4e^{-x^3}$$ I started taking the first, second, third, etc. derivatives but the expression for it seems to explode with terms. I was just ...
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### Determining the parts of an exponential equation based on the graph.

I am given the following equation and asked to find $a$ $$s(t) = Ae^{-at}.$$ I know $A = 9$ and I know at $t = 1, s(1) = 3$ so I create the following equation $$3 = 9e^{-a.1}$$ and reduce it down ...
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### Exponential Equation $p \cdot a^x - q \cdot b^x \le c$

I have a hard exponential equation to solve: Given $p,q,a,b,c$, I need to solve $x$ in the following equation: $p \cdot a^x - q \cdot b^x \le c$. Any suggestion is welcome. Thanks.
"A certain bacteria population is known to quadruple every $90$ minutes. Suppose that there are initially $120$ bacteria. What is the size of the population after $t$ hours?" I've been using this ...