Questions related to real and complex logarithms.

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3answers
34 views

Complex derivative involving exponents and natural log

Find: $\frac{d}{dx} a^{x\ln x}$ I have tried several methods involving u-substitution etc, but can't figure it out.
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1answer
17 views

Conversion of bases with logarithms

The question says if $\log_6(2)$ is $a$ and $\log_5(3)$ is $b$, express $\log_5(2)$ in terms of $a$ and $b$. I have tried the change of base formula for $ab$ to no avail, can someone give me a hint ...
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3answers
65 views

$(5^{2x}-1)(5^x)=1/5^x$ solve

I have the problem $(5^{2x}-1)(5^x) = 1/5^x$. I have already simplified it to $5^{3x}-1=1/5^x$ My question is when I do $\log$ base $5$ to the left side of the equation to get $3x-1$ by itself so ...
0
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1answer
58 views

Integration involving $\log_2(x)$

Having a hard time going about this problem: $$\int{\frac{\ln(2)\log_2(x)}{x}}$$ I believe $\ln(2)$ would be considered a constant, so than the equation would then changed to: ...
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1answer
41 views

Polytime implementation of Discrete Log using primitive recursive functions

The primitive recursive functions are defined by Godel as: $z() = 0$ $s(x) = x+1$ $\pi_i(x_1, \dots, x_k) = x_i$ Plus closure under Composition: $h(x_1, \dots, x_m) = f(g_1(x_1, \dots, x_m), ...
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4answers
157 views

How can $\frac12 \log(x) = \log(\sqrt{x})$?

How can $\frac{\log (x)}{2}= \log \left(\sqrt{x}\right) \\$? How would I come to this conclusion?
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2answers
52 views

Prove equality $a^{\log_b c} = c^{\log_b a}$

I'm try to prove the equality: $$a^{\log_b c} = c^{\log_b a}$$ I'm having trouble finding information regarding this, also I need to figure out why $n^{\log_2 3}$ is better than $3^{\log_2 n}$ as a ...
1
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1answer
29 views

Show that $g(x)=x\ln{x}$ and $g(x)=e^x$ are bounded below.

Show that $g(x)$ is bounded below, for $0\leq x$: a) $g(x) = \left\{ \begin{array}{ll} 0 & \mbox{if } x=0 \\ x\ln{x} & \mbox{if } x>0 \end{array} \right.$ b) $g(x)=e^x$ For (a), ...
0
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0answers
63 views

Complex exponentiation

So I've got this question that is a bit difficult to ask, since it uses a term in my language that I can't properly translate into English. For $z\in\mathbb{C}^*$ and $a\in\mathbb{C}$ it would be ...
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0answers
48 views

show that $(x^b+y^b)^{1/b} < (x^a+y^a)^{1/a}$ if $x>0$, $y>0$, and $0<a<b$

Show that $(x^b+y^b)^{1/b} < (x^a+y^a)^{1/a}$ if $x>0$, $y>0$,and $0<a<b$ by examining the sign of the derivative of an appropriate function. This is an exercise in middle part of ...
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3answers
219 views

“Linearize” an exponential-looking graph with log function

This may be a beginner question, but I can't quite wrap my head around logs... I have a set of data (from an experiment) which gives me an exponential-looking graph (Fig 1). I'd like to "linearize" ...
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1answer
30 views

Help with logarithmic equation?

Find the value of x if: $(2^x)^2 + 3(2^x) - 18 = 0$ So far, I have done $(2^x)^2 + 2^x(3)=18$ $(2^x)^2+2^x=6$ What should i do with $(2^x)^2+2^x$ so i can have only one $^X$ on the left side ...
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0answers
33 views

Help with finding $x$ value in logarithms?

Find the value of $x$ in: $2^{x+1} = 3^x$ So far I've done: $(x+1)\log_{10}{2}=x\ log_{10}{3}$ $x\log_{10}{2}+log_{10}{2}=x\log_{10}{3}$ What should I do next?
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1answer
23 views

Help with logarithms?

Find the value of $x$ in: $5.25 = -\log_{10} (x)$ What should I do with a negative log? Should i do $5.25= \frac{1}{\log_{10}(x)}$ ?
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2answers
42 views

Help finding value of x in logarithms?

How to find the value of x in: $$10=8.4\log(0.3x+1)$$ so far I got : $$10=\log(0.3x+1)^{8.4}$$ $$10^{10}=(0.3x+1)^{8.4}$$ What should I do next?
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5answers
104 views

Finding value of x in logarithms?

Q) Find the value of $x$ in $2 \log x + \log 5 = 2.69897$ So far I got: $$2 \log x + \log 5 = 2.69897$$ $$\Rightarrow \log x^2 + \log 5 = 2.69897 $$ $$\Rightarrow \log 5x^2 = 2.69897 $$ What ...
1
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3answers
105 views

$3^{2x} - 34(15^{x-1}) + 5^{2x} = 0$

I've never seen anything like this. Do somebody have a way to solve it? I've tried the basci exponential functions techniques but it does not work. Even substitution does not work... I'm really ...
0
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1answer
41 views

Converting base2 scientific notation to base10 scientific notation

I would now like to know how to convert a value in base2 scientific notation (is that the correct terminology?), say 1.93 * 2 ^ 88, into the form of A * 10 ^ B. I want to do this without expressing ...
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2answers
51 views

Find $n$ satisfying the equation $[\log_21]+[\log_22]+[\log_23]+\dots[\log_2n]=1538 $

If $[\cdot]$ denotes greatest integer function, then what is the value of natural number $n$ satisfying the equation $$[\log_21]+[\log_22]+[\log_23]+\dots[\log_2n]=1538 ?$$ My try: Note that ...
3
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2answers
90 views

Apply the natural logarithm fractional number of times

Let f_n(x) be the recursive function that adds 1 to x and takes the natural logarithm, ...
0
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2answers
98 views

How to prove $\sum\limits_{k=2}^{n}\dfrac{1}{k}<\log(n)<\sum\limits_{k=1}^{n-1}\dfrac{1}{k}$

How to prove $\sum\limits_{k=2}^{n}\dfrac{1}{k}<\log(n)<\sum\limits_{k=1}^{n-1}\dfrac{1}{k}$ It is clear if i consider the area under $f(x)=\dfrac{1}{x})$ from $1$ to $n$ end divide the ...
0
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1answer
23 views

Show that the following holds;

Let $h(p) = -p \log p-(1-p)\log (1-p)$ denote the binary entropy of a Bernoulli distribution when the probability of observing a zero is $p$, where $\log$ denotes the logarithm to base 2. Show, using ...
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2answers
41 views

proof of logarithmic property $\displaystyle a^{\log_{a}{b}}=b$

I don't know how to show that $\displaystyle a^{\log_{a}{b}}=b$ Can anyone give a hint?
27
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5answers
322 views

An integral with irrational exponents $\int_0^\infty\frac{\log\left(\frac{1+x^{4+\sqrt{15}}}{1+x^{2+\sqrt{3}}}\right)}{\left(1+x^2\right)\log x}dx$

I was challenged to prove this identity $$\int_0^\infty\frac{\log\left(\frac{1+x^{4+\sqrt{15\vphantom{\large A}}}}{1+x^{2+\sqrt{3\vphantom{\large A}}}}\right)}{\left(1+x^2\right)\log ...
0
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1answer
15 views

Creating a constrained log function

Good morning, I have a series of values that I intend to use as the exponents and I would like to create a log function so that: $Log_x(y_1)=.1$ $Log_x(y_2)=.2$ $Log_x(y_3)=.3$ ... ...
1
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1answer
33 views

How to prove this Logarithmic identity?

I am just learning logs, and can't get this one out ? How to prove that $$ \log_a (b) \cdot \log_b (c) = \log_a (c) $$ in the format : log [base]([argument]) ...
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2answers
32 views

Logarithmic Differentiation

When do we use : $ lnab = ln a + ln b $ and when do we use : $ \ln |y| = \ln |f_1(x)| + \ln |f_2(x)| + \cdots + \ln |f_n(x)| $ ? It is stated that we use the second form of log differentiation ...
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2answers
64 views

How to find the limit of $\lim_{n\to \infty} n(H(n) - \ln(n) - \gamma)$

How to find the following limit: $$\lim_{n\to \infty} n(H(n) - \ln(n) - \gamma)$$ where $H(n) = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ is the $n^{th}$ harmonic number and $\gamma$ is the Euler ...
0
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1answer
43 views

What is the same as the inverse of a logarithm?

I am trying to simplify $f(n) = \frac{n}{\log(n)}$ into a more easily understandable function. Up until now, I got as far as $n\cdot(\left(\log(n)\right)^{-1})$. Is there any way I can further ...
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2answers
66 views

solve equation with logarithm base 10

I am going back to study log and unfortunately I don't know a lot. I need to solve this: $$ 100= 10\log_{10} \left(50/x\right) $$ Actually is log base $10$, don't know how to format here. So I did ...
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1answer
98 views

Approximating Logs and Antilogs by hand

I have read through questions like Calculate logarithms by hand and and a section of the Feynman Lecture series which talks about calculation of logarithms. I have recognized neither of them useful ...
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3answers
26 views

Compare two powers

how can I compare these powers: $3^{500}$ and $5^{300}$ What I did is: $\log_3(3^{500})$ and $\log_3(5^{300})$ So I have $500$ and $\log_3(5^{300})$ Now I do not know what to do. Thank you in ...
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2answers
45 views

Help me prove the result of this limit: $\lim_{x\to \infty} {\log_{x^3+1} (x^2+1) \over \log_{2e^x+1} (x^2+5x)}$

The limit is $$\lim_{x\to \infty} {\log_{x^3+1} (x^2+1) \over \log_{2e^x+1} (x^2+5x)}$$ I know that it should be equal to $\infty$ but i have yet to prove it. Please help me do so.
2
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1answer
35 views

Definite integral of partial fractions?

So I'm to find the definite integral of a function which I'm to convert into partial fractions. $$\int_0^1 \frac{2}{2x^2+3x+1}\,dx$$ Converting to partial fractions I get... $\frac{A}{2x+1} + ...
1
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1answer
66 views

Derivative of $\operatorname{Log}(\operatorname{Log}(z^2))$

Please help me with this question: (i don't know how to start) Suppose that $f(z)$ = $\operatorname{Log}(\operatorname{Log}(z^2))$. Find $f'(z)$ where it exists, and determine the set of points at ...
4
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0answers
83 views

Differentiating $y=x^x$ with the formal definition of a derivative

A friend and I were messing around with derivatives, and while we both know the procedure for finding the derivative of $y=x^x$ with logarithmic differentiation, i.e. $$y=x^x\\ ln(y)=x*ln(x)\\ ...
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1answer
36 views

How solve this $\log { { x }^{ \log _{ x }{ y } } } =\quad \frac { 5 }{ 2 } \\ x+y=6\\ $

How solve this logarithm equation $\log { { x }^{ \log _{ x }{ y } } } =\quad \frac { 5 }{ 2 } \\ x+y=6\\ $
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1answer
70 views

Give the domain and range of $y=\log(x-3)+2$

I am so confused. I think the domain is $x>3$ but is the range ARN or is it $y>0$ . . .
2
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1answer
91 views

How to solve $x^{\log_3(x)} \geq \frac{1}{27}$

How to solve this? My problem is to solve: $$x^{\log_3(x-4)} \ge \frac{1}{27}.$$ The log base is $3$.
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1answer
65 views

How solve this logarithms equation

What relationship between a,b and c ?
3
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1answer
57 views

Logarithms, prove this limit.

Mathematica knows that: $$\log (n)=\lim_{s\to 1} \, \left(1-\frac{1}{n^{s-1}}\right) \zeta (s)$$ Kind of tautological starting with logarithms, but I would like to know better why this limit works: ...
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3answers
39 views

How to move from powers to simple logarithms?

I'm following a book that briefly moves from $$16000 \times 2^{\displaystyle \left (-\frac{x}{24} \right )} = 1600$$ to $$x = \frac{24 (\log(2) + \log(5))}{\log(2)}$$ adding the comments that ...
6
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1answer
103 views

How to compute the mean average exponent of the naturals? What is the limit for large numbers?

With a friend I was trying to get an understanding for why the expected gap between primes is logarithmic. With that motivation I tried to express the average exponent of numbers. By average ...
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1answer
85 views

Linear to semi-logarithmic scale

I've got some FFT results I want to draw with a log10 scale on the x axis. Let's call nBins the number of bins (window size / 2) nPixels the total number of pixels We will assume that the ...
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2answers
31 views

How is natural log integration broken up into this range? (equation is contained the script)

When I was reading a paper, I found an strange derivation like $$\int^{+\infty}_{-\infty}\mathrm{ln}(1+e^w)f(w)dw\\=\int^0_{-\infty}\ln(1+e^w)f(w)+\int^\infty_0[\ln(1+e^{-w})+w]f(w)dw$$ when $w$ is ...
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2answers
78 views

Evaluate $\frac{d}{dx}\{(\sin x)^{\cos x} + (\cos x) ^{\sin x}\}$ with logarithmic differentiation

Spivak asks us to evaluate $$\dfrac{d}{dx}\{(\sin x)^{\cos x} + (\cos x) ^{\sin x}\}$$ by logarithmic differentiation. Does he mean for us to evaluate each term separately (which seems to turn out to ...
4
votes
6answers
107 views

$\lim_{n\to\infty}\left(1+\frac{3}{n}\right)^\frac{n}{2}$

I am trying to resolve this to number $e$. However, I would like to do it in the simplest form. just a note I already tried wolfram but I would like someone to give me a simpler solution. ...
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votes
2answers
115 views

Prove that $e ^ π$ > $π ^ e$. [duplicate]

Prove that: $$e ^ π > π ^ e.$$ Hint: Take the natural log of both sides and try to define a suitable function that has the essential properties that yields the above inequality
10
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2answers
264 views

$\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$

Prove that: $\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$, for $n > 1,\, n\in \mathbb{N}$ For example. For $n=2$, we have $\lfloor ...
0
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0answers
30 views

Why can't the base of a logarithm be negative? [duplicate]

I understand why the base of a logarithm can't be 0 or 1, but why negative? What I found out is that when the base is negative we get imaginary results when the powers are rational numbers with odd ...