# Tagged Questions

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### Proving $\log(b^a) = a \log(b)$ using calculus

Sorry, this is a really simple question, but I'm trying to teach myself calculus and can't figure it out. If we define $\log(b) = \frac{db^x}{dx}(0)$ how does one prove $\log(b^a) = a\log(b)$? I ...
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### Exponential function to logarithmic function

i'm stuck on completing this equations. Is this correct? $$z=a e^{-bt}$$ $$\ln(z)=\ln(a)+\ln(e^{-bt})$$ $$\ln(z)=\ln(a)+(1)(-bt)$$ $$\ln(z)=\ln(a)-bt$$
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### Choosing a branch of the square root

Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With ...
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### Consumption change calculation

I want to calculate yearly consumption change according to the following formula: $$C_{t+1}=C_{t}e^{x_{t}}$$ I need to calculate ${x_{t}}$. I have the consumption data $C_{t+1}$ and $C_{t}$.
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### How can I solve the equation…

$$\log\left(\frac{\pi_i}{1-\pi_i}\right)=\sum_{k=0}^K x_{ik}\beta_k\qquad i=1,2,\dots,N$$ How can I make the equation above the one below by taking "$e$" to both sides. Note that after taking $e$ ...
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### Noncircular construction of $e$ and $\ln$ for the real line

Could anyone direct me to (or possibly detail) a construction of $e$ and $\ln$ along the reals? For example, they can define $e=\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$ but from this definition ...
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### Why is $3^n$ not in $\Theta(2^n)$

How is it that $3^n$ not in $\Theta(2^n)$, while $log_3 n$ is in $\Theta(log_2 n)$ ?
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### How to solve this equation with linear n as well as polynomial n?

I am banging my head against the wall, but somehow I can't find a closed form solution to this equation in n: $$229,244 + 58,044 \cdot n = 130,000 * 1.78^n$$ Obviously, if there was no $n$ ...
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### Calculating an exponentially increasing vector of points in a test and measure system

My application is setting and measuring current and voltage in a physical system with a software algorithm. Given these parameters: min, ...