Questions related to real and complex logarithms.

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-2
votes
3answers
48 views

Imaginary part of $ln(\sqrt{i})?$ [on hold]

Which of the following is the imaginary part of a possible value of $\ln(\sqrt{i})?$ (a) $\pi$ (b) $\pi/2$ (c) $\pi/4$ (d) $\pi/8$ I compute $\sqrt{i}=\dfrac{1+i}{\sqrt{2}}$, but how to proceed ...
1
vote
2answers
17 views

Find the general expression from the antiderivative

I am having trouble computing the original function. Question states: Let $f$ be a differentiable, positive function, such that $$f'(x)=x*f(x)$$ for all real numbers x. A) Find the general ...
0
votes
1answer
13 views

Confusion: dB to scalar and from scalar to dB

Assume we have $$N_1=5 \text{ dB}$$ $$N_2= - 110 \text{ dB}$$ Then we have $$Y=N_1+N_2=-105 \text{ dB} $$ If I convert to scalar then $$10 \log(X) = -105 \rightarrow X=10^{-10.5}$$ Let me start the ...
1
vote
1answer
37 views

Solving a ln divided by a ln.

I am having trouble figuring out how to calculate this. Thank you for your help. $$.926 = \frac{ln(1+.8u)}{ln(1+u)}$$ What does $u$ equal?
0
votes
0answers
10 views

Standard deviation errors in log scale

I have a not so common issue with error bars in the log log scale. To be more precise, I have measurements of a quantity Y with an associated standard error Yer that has normal distribution and these ...
1
vote
1answer
28 views

How do you simplify this logarithm?

$$\large\log\sqrt[3]{\frac{x^2y^5}{z}}$$ I think this is the answer, but I'm not positive:$$\frac{1}{3}\left((2\log{x}+5\log{y})-(\log{z})\right)$$
0
votes
1answer
25 views

Need help with understanding manipulations on logarithms

I cannot understand the result from logarithms manipulations even though I am going over logarithmic properties. I am simply stuck. So here is the problem: $$n = 2^k \implies k = \log_2n$$ $$x = 3^k ...
2
votes
1answer
49 views

How to show without calculator that $\left\lfloor\, \log_{10}{999^{999}}\right\rfloor =\left\lfloor\, \log_{10}{999^{999}}+\log_{10}2\right\rfloor$

By wolfram alpha, I get $\left\lfloor\, \log_{10}{999^{999}}\right\rfloor =\left\lfloor\, \log_{10}{999^{999}}+\log_{10}2\right\rfloor=2996$. How to prove that $\left\lfloor\, ...
0
votes
1answer
57 views

Why is -ln x is not equal to 1/ln x?

I am doing differential equation now and I need to convert them into the proper form in order to do my homogeneous differential equation. So now I just found out that -ln x is not equal to 1 / ln x. I ...
0
votes
1answer
25 views

Solutions to $N=2^r-r-1$

Considering this equation $N=2^r-r-1, N \in \mathbb{N},r \in \mathbb{N}$. I've tried (unsuccesfully) working out $r$ given $N$. However I recently stumbled across a website which was using $\left ...
0
votes
1answer
16 views

Unique intersection of $b^x$ and $\log_b(x)$

It seems to me that there is exactly one real number $b>1$ such that the graphs of $y=b^x$ and $y=\log_b(x)$ intersect at a single point. What exactly is this number?
1
vote
3answers
58 views

Limit of logarithms exponential

$$ \lim_{x\to\infty}\biggl(\frac{\ln(x-2)}{\ln(x-1)}\biggr)^{x\ln x}. $$ L'Hopital seems like a very hardcore solutions given the situation.Are the any other options?
1
vote
7answers
54 views

Logarithms with an answer that is a fraction

How does log base $16$ of $32$ equal $1.25$? If we divide $32/16=2$ but then if we divide $2/16$ it doesn't come out to a whole number unlike with log base $2$ of $4$ where $4/2=2$ and $2/2=1$ I am ...
-1
votes
1answer
27 views

How to show the following inverse trigonometric equation? [closed]

Let $a$ and $b$ be real numbers so that $ b\neq 0 $. Show that $ \dfrac{tan^{-1}\left( \dfrac{a}{b}\right)}{\pi}=\dfrac{\ln \left( b/ \sqrt{a^{2}+b^{2}}+ai/ \sqrt{a^{2}+b^{2}}\right) }{\ln (-1)}$.
0
votes
2answers
24 views

Simplifying the expression. [closed]

Can anyone help with this? You invest $£ 4,624$ in a saver account with an interest rate of $0.42$ percent compounded monthly. After how many months does the balance on you account reach $£ ...
2
votes
0answers
17 views

Taking logs multiple times

Is there a formule to calculate (log (log ( log ... log n))) assume all the base to be the same (b)? I was not able to find one on wikipedia.
0
votes
2answers
42 views

Simplifying the exponential expression $e^{-4\ln x +8\ln y +2}$ [closed]

I'm totally stuck on this. Tried numerous sites for a decent explanation but can't find anything. Simplify the expression $$e^{-4\ln x +8\ln y +2}.$$ Thanks in advance.
0
votes
1answer
23 views

What is the difference between log scale. and plotting logarithms?

I find plots in scientific literature beyond confusing. I understand quite clearly the difference between a linear and a logarithmic scale, and when each is desirable. Suppose we are plotting values ...
0
votes
2answers
33 views

Solve for $m$ in $d^m = n$ [duplicate]

I believe the answer is $m = \lceil \sqrt[d]n \rceil$ or $\lfloor \sqrt[d]n \rfloor$. Can anyone help me?
-2
votes
0answers
43 views

Logarithms-Solve for x.

$4^{2x}=3^{x-1}+5$ Solve for x. I tried: $\log 4^{2x}=\log 3^{x-1}+\log 100000$ $4^{2x}=[3^{x-1}]100000$
0
votes
1answer
74 views

Logarithms of Negative Numbers

In Algebra II, I learned that you cannot take the logarithm of a negative number. However, when visiting the topic again, I realized that the identity $e^{i \theta} = \cos{\theta} + i\sin{\theta}$ ...
0
votes
1answer
15 views

Computing Principal Logarithm on Different Intervals

Compute the principal logarithm of a complex number $z=\sqrt{3}+i$ using $\mathrm{Arg}(z) \in [0,2\pi)$ and $\mathrm{Arg}(z) \in (-\pi,\pi]$. Wikipedia shows how the answer can be different for the ...
0
votes
1answer
18 views

Laws of Logarithms

log 1050 = log 73.3 + 0.75 log m I tried this: log 1050 = log 73.3 m^0.75 1050/73.3=73.3(m^0.75)/73.3 m^0.75=14.3247 logm14.3247=0.75 How do you solve for m?
2
votes
0answers
42 views

Is the product rule for logarithms an if-and-only-if statement?

If a function $f(x)$ is proportional to $\ln x$, then we know $$ f(xy) = f(x) + f(y). $$ My question is, Is the converse true? If we know that, for an unknown function f, $$ f(xy) = f(x) + f(y), $$ ...
1
vote
0answers
9 views

Tigher bounds on concavity of log

Is there a tighter upper and lower bound on concavity of $\log(.)$ function? It is very well known that $\log(\sum_{i}p_ix_i) - \sum_{i}p_i\log(x_i) \geq 0$. But are there stronger versions of this ...
1
vote
2answers
95 views

How to find the limit $\lim_{x\to\infty}\frac{\ln(x^2+x+5)}{\ln(x^8-x+3)}$?

I don't know how to approach this problem: $$\lim_{x\to\infty}\frac{\ln(x^2+x+5)}{\ln(x^8-x+3)}$$ I tried reducing it to the form $\lim (1+x_n)^{1/x_n}=1$, but that didn't work.
2
votes
1answer
80 views

Equation $e^{\frac{1}{x}} - x =0$

Can someone solve this equations with steps $$e^{\frac{1}{x}} - x =0$$ I dont know how to start. I tried adding logarithms but that doesn't help.
1
vote
1answer
43 views

Let $f(x) = \exp (x^2 − x + 6)$. Choose Dom(f) so that $f^{−1}$ exists. What is $f^{−1}$ and Dom($f^{−1}$) in your case?

I have already got $$y=\exp(x^2-x+16)$$ $$\ln y = x^2-x+6$$ $$\ln x=y^2-y+6$$ I know for getting inverse function we need to solve for $x$, but what should i do in this case?
0
votes
0answers
20 views

Trying to understand how the trapezoidal rule applies to a derivation of Stirling's Approximation

I am reading through the wikipedia article on how to derive the Stirling's Approximation. The article applies the Trapezoidal Rule to get the following: $$\begin{align} \ln (n!) - ...
-2
votes
2answers
44 views

Solving exponential equations like $6^{3x}=4^{2x-3}$ using logarithms

I'm trying to solve these using logarithms: $a$) $9^{x+1} = 27^{2x-3}$ $b$) $6^{3x} =4^{2x-3}$ $c$) $210=40(1.5)^x.$ I'm trying to practice logarithms by doing various questions. It's been a ...
0
votes
1answer
25 views

Logarithms and exponential decay

The table describes the cooling of a cup of coffee as it sits on your teacher’s desk in the math office. Time (min) $0, 4, 8, 12, 16, 20$ Temperature (celsius) $55, 47, 40, 34, 29, 25$ a) Calculate ...
0
votes
2answers
23 views

What is the acute angle $A$ if $\log_4(\sin^2A)=-1$?

If angle $A$ is acute and $\log_4(\sin^2A)=-1$, then the value of $A$, to the nearest tenth of a radian, is... I've worked through this some different ways but I haven't made any progress. Please ...
0
votes
0answers
10 views

Exponential decay of the temperature of coffee

The table describes the cooling of a cup of coffee as it sits on your teacher’s desk in the math office. Time (min) 0 4 8 12 16 20 Temperature (celsius) 55 47 40 34 29 25 a) Calculate a, the ...
-1
votes
2answers
32 views

Forming equations for exponential growth/decay questions [closed]

Dry cleaners use a cleaning fluid that is purified by evaporation and condensation after each cleaning cycle. Every time the fluid is purified, 2.1% of it is lost. The fluid has to be topped up when ...
1
vote
2answers
66 views

How can I prove that $x-{x^2\over2}<\ln(1+x)$

How can I prove that $$\displaystyle x-\frac {x^2} 2 < \ln(1+x)$$ for any $x>0$ I think it's somehow related to Taylor expansion of natural logarithm, when: $$\displaystyle ...
0
votes
1answer
26 views

Condensing logarithmic expressions

See attached images. Not sure where I'm going wrong.
12
votes
0answers
96 views

$\log_2 13$ is irrational.

$\log_2 13$ is irrational. Is it true? $x=\log_2 13$ $\implies 2^x=13$ So, it will be an irrational number, if not,$$x=\frac p q$$ and $$2^{\frac p q}=13$$ $$\implies 2^p=13^{q}$$ Since, $13$ is ...
1
vote
1answer
39 views

Finding original function of $1/x$ by using a step function

Is it technically possible to show, that the area under $1/x$ in the interval $[1,a]$ equals $\log a$ without using any kind of differentiation, but only step functions. I tried this, but without ...
1
vote
2answers
42 views

What is the behavior of these functions linear or logarithmic or neither?

Please consider the following functions $F$ and $G$. \begin{align*} F(K) = \log_2 \left(\frac{( 2\sqrt{K-1}+K-2)^2}{(\sqrt{K-1}-1)^2}\right)+(K-1) \log_2(2) \end{align*} and the function ...
0
votes
3answers
28 views

Solve for x in log question?

If $2^x$ (2 to the power of x) $= 100$, what is $x$? I got $100/\log2$. Is that correct? I know how I solved it but now I don't get how I did and why I did what I did. The choices were... $$2 / ...
0
votes
1answer
44 views

How do I prove that the only possible function is $exp$?

Let´s say we have a differentiable function $f : \mathbb{R} -> \mathbb{R}$ with $f' = f$ and $f(0) = 1$ . How do I show that the only possible function for this to work $f = exp$ ? ...
0
votes
1answer
30 views

If $m \ge 8 s \log(m^2 s)$, how much greater $m$ is relatively to $s$?

Given that $m,s \in \mathbb{N}$, if $m \ge 8 s \log(m^2 s)$, how much greater $m$ is relatively to $s$ ? It seems to me $m>>s$, but I would like some idea of the magniture. I'm not quite sure ...
2
votes
0answers
71 views

Solve for $x: \ln(x+4)+\ln(x-2)=5$

Solve for x: $\ln(x+4)+\ln(x-2)=5$ Where do I go from here? If there weren't four terms in the equation I would use the quadratic formula. How can I solve for x? EDIT 1: Is this correct? ...
-2
votes
1answer
33 views

How to define human-friendly axis marking notches

I have a need to compute values of "tick" marks on an axis that uses various different min/max ranges. Specific example -- Picking Human-Friendly Notch Step Axis Y has function values range between ...
3
votes
1answer
62 views

Solving $\log_2(x^2) = x$ explicitly?

I'm having problems getting a proper step-by-step solution to this equation. $$ \log_2(x^2) = x $$ I know the results are 2 and 4, but so far I can get only solutions like these: $$ 2^x = x^2 \qquad ...
-1
votes
1answer
40 views

Is this logarithmic inequality true?

Assume we have two complex variables $h_i$ and $h_d$ which satisfy the following relationship $$ 2\ |h_i|^2\leq \ |h_d|^2$$ can we say that $$\log\left( 1+ \frac{\big||h_d| - ...
4
votes
1answer
295 views

inequality $10<2^{2^{\frac {3}{\log_2 \log_2 10}}}$

While working on this question I ended up with $10<2^2{^{\frac {3}{\log_2 \log_2 10}}}$ I am looking for answers using methods similar to this or this or this or this. Alternative original ...
0
votes
0answers
34 views

Determine if the graph $f(x) = \ln(x)$ has any critical numbers

Determine if the graph $f(x) = \ln(x)$ has any critical numbers. The derivative would be $f'(x) = \frac{1}{x}$
1
vote
0answers
27 views

Logarithmic Series [duplicate]

I was doing a bit of math when I came across logarithmic series. I have no idea from where they come from. They seem so unrelated, that I have no intuition behind them at all. So, can anyone prove ...
0
votes
1answer
53 views

question about logarithms [closed]

Is it true that for $n\ge 2$ integer and $x\ge 1$ real, we have $\lfloor \log_nx \rfloor=\lfloor\log_n \lfloor x \rfloor \rfloor$ ?