# Tagged Questions

Questions related to real and complex logarithms.

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### Prob. 7, Chap. 1 in Baby Rudin

Here's problem 7 in the exercises following Chap. 1 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Fix $b > 1$, $y > 0$, and prove that there is a unique real number $x$...
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### Show that $f(x^a) = a f(x)$ and $f(x y) = f(x) + f(y)$

Let $f(x) = (x-1) \large\prod_\limits{n=1}^{\infty} \dfrac{2}{x^{2^{-n}} + 1}$ For real $x > 0$ it is easy to show that $f(x^2) = 2 f(x)$. Let $a$ be a real number. Question 1 Show that ...
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### Solve for $x$ in $\frac{x}{\ln(x)}=a$. Why does Wolfram alpha use complex numbers here?

Is there any possible way of doing this without using complex numbers? And why are complex numbers used?
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### Find the value of $\log_{xz}(xy^4z) \times \log_{xy}(xyz^4)$

if $x,y,z \gt 1$ and $x^2=yz$ find the value of $$E=\log_{xz}(xy^4z) \times \log_{xy}(xyz^4)$$ what i did is $$E=\log_{xz}(xy^4z) \times \log_{xy}(xyz^4)=(1+4\log_{xz}y)\times (1+4\log_{xy}z)$$ ...
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### Change of base proof without reciprocal

I am looking for a proof of the change of base formula without using the reciprocal. I know that: $$log_ax=\frac{log_bx}{log_ba}$$ The proof usually involves taking the reciprocal: $log_ax=y$ ...
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### A group with unusual discrete log properties.

Does there exist a group where computing $g^x$ from $g^{a^{x}}$ is easy, computing $g^{a^{x}}$ from $x$ and $g^{a}$ is hard, and computing $x$ from $g^a$ and $g^{a^x}$ is hard. Intuitively I would ...
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### Solving logarithmic equations without calculator

Hi I am stuck on this question $$\log_x 10= 5 (\log_{10} x) +4$$ The answer key gives the solutions $x = 10^{1/5}$ and $x = 1/10$.
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### Find the value of $3^{\log_4(5)} - 5^{\log_4(3)}$. [closed]

Find the value of $3^{\log_4(5)} - 5^{\log_4(3)}$. Is there any property that can help here?
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### Divergent function of ratio must be logarithm

Given. Consider two functions $F(t)$ and $r(t,x)$ such that $\lim_{t\to\infty} F(t) = \infty$ and $\lim_{t\to\infty} r(t,x)$ is finite for any $x$. ($x$ and $t$ are always positive in what follows.) ...
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### Paradox: Summation of natural logarithms

Consider the expression : $$\sum_{i=1}^{\infty}\ln(i+2)-\ln(i+4)$$ If one evaluates it out, one gets $$\ln(\frac{3\times4\times5\times6\times...}{5\times6\times7\times8\times...})=\ln(12)$$ That ...
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### Why does $\ln(1+\frac{3}{n^2}+o(\frac{1}{n^2}))=\frac{3}{n^2}+o(\frac{1}{n^2})$?

In order to show that a series converges, I want to show that $\sum\ln(\frac{v_{n+1}}{v_n})$ Which led me to the following first part of the equation, but I didn't achieved to solve it so I looked in ...
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### I would like to calculate this limit: $\lim_{n \to \infty}(n^2+1)\cdot(\ln(n^2-4)-2\ln(n))$

I would like to calculate this limit: $$\lim_{n \to \infty}(n^2+1)\cdot(\ln(n^2-4)-2\ln(n))$$ but I am a bit lost on how to tackle the logarithm. Any help would be greatly appreciated.
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### Is $\log(n!) \in\Theta(n \log n)$ [duplicate]

Is $\log(n!) \in\Theta(n \log n)$? I know it is $O(n \log n)$ because $\log(n!) \leq \log(n^n)$ which is the same as $\log(n!) \leq n \log n$. But how can I show it is also $\Omega(n \log n)$?
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### logarithms properties

I know it's very easy and naive, but apparently I cannot understand the following equation. Can you please prove it? Thanks in advance. The equation is: $$2=3^{\frac{\ln{2}}{\ln{3}}}$$
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### How do you solve the equation $0.5^x = 2^x + 3$?

I need help with the following problem: $0.5^x = 2^x + 3$ I know the answer is -1.72, but I have to explain step by step how to solve it and I'm not sure how. I know you're supposed to take the log ...
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### Which is asymptotically larger $n^2 \log(n)$ or $n (\log(n))^{10}$?

Which is asymptotically larger $n^2 \log(n)$ or $n ( \log(n))^{10}$? I have tried by plugging in the values and $n^2 \log(n)$ turns out to be bigger. How can this be done analytically?
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### Can I take an exponent out of a sum?

For example, assuming we had a sum: $$\sum_{n=1}^m n^b \quad m,b\in\mathbb{N}$$ Is there any way to take the $b$ out of the sum? I tried taking the $\log_n$ of every value, add them together then ...
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### A $\log \Gamma$ identity: Where does it come from?

$$\log \Gamma (n)=n\log n -n +\frac{1}{2} \log \frac{2\pi}{n}+\int_0^\infty \frac{2\arctan (\frac{x}{n})}{e^{2\pi x}-1} \,\mathrm{d}x$$ Is an identity that is derived from using Sterling's ...
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### Solving logarithmic equation, different bases

What number do I need to multiply both sides with? I have worked for an hour on this but it is the first time I am using this website so it is impossible for me to write what I have already done. If ...
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### Integrate $\int(\log(\sin x \cos x))^n dx$ with hypergeometric function form

Evaluate $$\int({\log(\sin x\cos x)})^{n} \, \mathrm{d}x$$ with result in hypergeometric function form Could anyone help me with that?
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### Find number of digits of a number in another base

How can I solve this question: Given that $\log 3$ is about $0.48$, approximately how many digits are in the number $10^{150}$ if it were written in base $3$. Thanks!
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### Finding Stationary Points of Natural Log Function

$$f(x) = x - 2\ln(x^2 + 3)$$ I started by using the chain rule on $x^2 + 3$ which gives me $\frac{2x}{x^2} + 3$. At this point I tried to multiple $\frac{2x}{x^2} + 3$ by $x - 2$ - is this correct?...
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### Rejecting a solution.

Why does it for $x^2=9$ we get two solutions, while if we use the "log both sides" property the negative solution is rejected? which method is true and why?
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### Logarithmic equation with variable both “free” and in logarithm

I am trying to calculate an area bordered by two functions and in the process I need to solve this equation: $$e^{-10x}=-2x+1$$ I make it into a non-exponential form: $$-10x=ln(-2x+1)$$ And now I am ...
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### integral involving error function (erf)

Does anybody know if a closed form of this integral exist? $\int \mbox{erf}(x) \ln(\mbox{erf}(x)) \Bbb dx$ where erf is so called error function. In case there is no closed form solution. Is it ...
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### Derivation for Napier Logarithm

Something new that I came across called the Napier Logarithm. I was reading the book “Computing: A Historical and Technical Perspective”, and the book says that $$Nap.log x = 10^7(\log_e(10^7/x))$$ I ...
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### What is the integral of log(z) over the unit circle?

I tried three times, but in the end I am concluding that it equals infinity, after parametrizing, making a substitution and integrating directly (since the Residue Theorem is not applicable, because ...