Questions related to real and complex logarithms.

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-1
votes
5answers
69 views

Squaring a logarithm when the base is a square root

How is this equality obtained? $$ -\log_\frac 1 {\sqrt 2}(x - 7) = \log_2 (x - 7)^2 $$ I understand the process until this point $$ \log _\sqrt 2 (x-7) . $$ How do I get from there to $$ \...
0
votes
2answers
32 views

Problem solving with mass in terms of logs and exponentials

I've just been accepted to take my PHD in chemical engineering in Melbourne next year. Some how I have gone from the age of 17 with out taking too many extra maths classes and so at the moment (I'm 26)...
1
vote
2answers
101 views

Avoiding subtraction for finite difference with log and exp

I want to approximate the derivative of f(x) Finite difference $f'(x) \approx \frac{f(x+h)-f(x)}{h}$ I was taught that the error from the subtraction is blown up for small h. This I can verify with ...
12
votes
3answers
867 views

Integral involving logarithm: $\int_0^\infty \frac{ \ln x}{(x+a)(x+b)} dx$

How to solve the following integral $$\int_{0}^{\infty} \frac{ \ln x}{(x+a)(x+b)} dx,$$ where $a,b>0$ and $a \neq b$. I was looking for some kind of substitution. However, I don't see an obvious ...
9
votes
2answers
2k views

Why do these “equal” logarithms give different answers

This came across a discussion amongst Algebra 2 teachers at my school. We know $a\log x= \log x^a$ Say $2\log x=5$ $\log x^2 =5$ When $\log x=\log_{10} x$ Solving the first equation yields $x=10^{5/2}$...
1
vote
1answer
50 views

How to find exact length of digits or number of digits of $a^b$?

If $a$ and $b$ are positive integer then what is length of digits of $a^b$? I have worked so far and formula works fine. To find the exact length of digits of $a^b$ where $a\gt 0, b\gt 0$: Number ...
0
votes
2answers
44 views

Finding the value of x, logarithms and exponentials

I'm working through some logs and exponentials questions at the moment in order so that I might be a little prepared for any I might utilize in a science PHD. I'm currently getting through the ...
1
vote
4answers
25 views

Half-life of Am-$241$, $3$ micrograms decays over $9$ years, how much if left?

$3$ micrograms of Americium-$241$, which has a half life of $432$ years. After $9$ years how much will remain? I'm not sure of the formula to use or how to calculate it. I'm assuming it's exponential ...
-2
votes
5answers
97 views

Prove $\ln^2(x)>\ln(x+1)\cdot\ln(x-1)$ for $x>2$ [closed]

Could anybody please help prove the following: $\ln^2(x)>\ln(x+1)\cdot\ln(x-1)$, for $x>2$.
-2
votes
1answer
39 views

Does, S = k ln W == W = e^s/k? [closed]

"Boltzmann's equation relates the entropy S of an ideal gas to the number W of microstates corresponding to a given macrostate, via the equation S = k ln W where k is the so-called Boltzmann ...
0
votes
1answer
35 views

Newton's Law of Cooling (and Heating)

The Formula for the equation is as follows: $$T(t)=\frac {\int^t(−T_s)ke^{-kt'}dt'+C}{e^{-kt}}$$ This formula is needed to determine the temperature at time $t$, $T(t)$, of an object as it begins to ...
1
vote
0answers
19 views

Interpolation / point fitting onto a logarithmic line segment

I have figure which is logarithmic scale on both axis. There's a line on that figure, I know two points on that line and want to interpolate a third point on that line based on the two known points. ...
1
vote
1answer
25 views

Newtons Law of Cooling in Forensic Science

Question goes: Law enforcement would like to know the time at which a person died. The investigator arrived on the scene at 8:15pm, which we will call $t$ hours after death. At 8:15 (i.e $t$ hours ...
0
votes
1answer
45 views

Logarithm Rules Ambiguity

I'm having some problems explaining myself the following ambiguity. According to logarithm rules: $\ln6=\ln(2\cdot3)=\color\red{\ln2+\ln3}$ $\ln6=\ln((-2)\cdot(-3))=\ln(-2)+\ln(-3)=\color\red{\...
0
votes
1answer
45 views

Logarithm problem question

$$a^{bx} = c$$ Solve for x $$\log a^{bx} = \log c$$ $$bx \log a = \log c$$ $$x = \frac{\log c}{b \log a}$$ Is this correct? Thanks :)
1
vote
5answers
68 views

Solve for $x$ : $\log_e(x^2-16)\lt \log_e(4x-11)$

$\log_e(x^2-16)\lt \log_e(4x-11)$ My attempt: Since the base is $\gt 1$, we have from the above , $$x^2-16-4x+11\lt 0\\ \implies x^2-4x-5\lt 0\\ \implies(x-5)\cdot (x+1)\lt 0$$ If I say $(x-5)\gt ...
2
votes
2answers
32 views

Questions about Exponentiation and roots and logarithms.

in this page a few questions I want to ask you about the Exponentiation and roots and logarithms: What and how the Exponentiation definition can be defined by real numbers.? What is the overall ...
0
votes
2answers
37 views

Why is my answer incorrect for this differentiation question?

$$y = x* ((x^2+1)^{1/2})$$ I must find $$dy/dx$$ $$u = x, v = (x^2+1)^{1/2}$$ To do this I must use the product rule and the chain rule. To get dv/dx, $$(dv/dx) = (1/2)*(b)^{-1/2}*2x $$ $$(dv/dx) ...
0
votes
1answer
29 views

derive the pdf for “difference of log-normal distributions”

Can someone please help me to derive pdf for $X$, $$ X = \frac{\ln(f_1) - \ln(f_2)}{b_2-b_1} $$ here $f_1$ and $f_2$ are normal distributions with different means and standard deviations, and $b_1$ ...
0
votes
1answer
25 views

Log power rule problem

According to many parts of the Internet, this log rule is used. log(a^b) = b*log(a) The proof is: Now let's say I want to use the rule in a Cartesian ...
0
votes
2answers
40 views

Linear equation from log equation

Further mathematics is driving crazy at the moment as I prepare for a PHD in chem eng. I've been working hard at the books but this one has caught me out. I basically need to derive a linear equation....
0
votes
1answer
39 views

Finding x and y from two given logarithmic equations

I'm just studying some further mathematics units for my own benefit before I undertake a PHD in chemical engineering next year. I feel the learning of the mathematical concepts at this level will help ...
2
votes
5answers
227 views

Is the natural logarithm actually unique as a multiplier?

The Wikipedia page on the natural logarithm says: 'Logarithms can be defined to any positive base other than 1, not only e. However, logarithms in other bases differ only by a constant multiplier from ...
1
vote
1answer
20 views

$\lim_{x\to 0}x^a\log^k(x)$ where $a>0,k\in\mathbb N_0$

I'd know how to solve this for $k=0$ or $k=1$ for example, but I'm currently lost trying to prove the limit is zero for any non-negative integer $k$. I'd appreciate any hints!
2
votes
1answer
47 views

Why does $\ln x / \ln b = \log_b x$?

I'm doing some Java code. As far as I can tell, Java only has functions that do natural log and base $10$ log. I have a requirement to specify the base. I've seen that doing $\ln x/ \ln b$ is the ...
0
votes
1answer
35 views

Proving logarithmic maths graphically

I'm just going through some further maths units as I prepare for a PHD in chemical engineering. I'm finding the thought processes to be invaluable in my problem solving skills. However, I recently ...
5
votes
1answer
14k views

Why the number e(=2.71828) was chosen as the natural base for logarithm functions? [duplicate]

Possible Duplicate: What's so “natural” about the base of natural logarithms? Why the number e(=2.71828) was chosen as the natural base for ...
62
votes
7answers
3k views

What's so “natural” about the base of natural logarithms?

There are so many available bases. Why is the strange number $e$ preferred over all else? Of course one could integrate $\frac{1}x$ and see this. But is there more to the story?
15
votes
3answers
2k views

Why is Euler's number used as a base for logarithms? [duplicate]

Is there some special property of '$e$' which makes it suitable to be used as a base for logarithms? Moreover, does the natural logarithm possess some advantage over the common logarithm? I don't ...
0
votes
2answers
18 views

Solving equation of form $x = -a/ln(bx)$

I have an equation that I am trying to solve, which can be reduced to the form $$ x = -\frac{a}{\ln(bx)}$$ where I am trying to solve for $x$. Mathematica says the solution is of the form $$x = \...
0
votes
3answers
22 views

Need to overcome erroneous result when differentiating natural log of a fraction

I am trying to differentiate the following: $$ln(3x-8/6x+2)$$ my (incorrect) method is: let $$ln(x) = ln(u)$$ therefore when differentiating u.. $$ln(u) = 1/u$$ and diff of$$$$(3x-8/6x+2) = 3/6 = 0....
2
votes
2answers
40 views

Evaluate $\lim_{x\rightarrow\infty} \ln(x^2+1)-\ln(x-1)$

How would you solve the following limit? The method I used can be seen below. I'm just not sure if it's valid. I was thinking perhaps a substitution for $(x-1)$ might also work, but when I followed ...
0
votes
1answer
23 views

Algebra, rewriting a formula

I have to rewrite this formula: $$10^{-5,6-0,4m}=\frac{c}{x^2}$$ To: $$m(x)= -14,0-2,5logc + 5,0logx$$ But im stuck at: $$m(x)= \frac{2logx -logc+5,6}{-0,4}$$ and have no idea how to continue from ...
0
votes
1answer
29 views

Question regarding integration of $ln(f(x)^{g(x)})$

I am trying to solve the following integral: $$\int ln[(x+2)^{x+5}] dx$$ I'm not entirely sure how to go about this. Since I know that $\int udv = uv - \int vdu$ I started by assigning $u = ln[(x+...
0
votes
0answers
6 views

Equation of semilog line

I am trying to determine the equation for a line passing through points (10, 0.5) and (100,0.3) where the x axis is on a logarithmic scale and the y axis is on a regular scale. This should be simple ...
1
vote
0answers
19 views

Issue an integral involving a $\log$

Let $$F(t)=\frac{t+7}{2+t}$$ and $$E(t)=\frac{\ln(t+4)}{t+2}\,.$$ My job is to compute the area between them from $x=0$ to $x=5$, which got me from $$\int_0^5{F(t)-E(t)}$$ to $$\int_0^5{\frac{t+7-\ln(...
1
vote
1answer
41 views

Natural Logs and Anit-Derivatives are kicking me

I am given a problem involving rates of flow, $F(t)=\frac{t+7}{2+t}$ is the rate at which a bucket is being filled. The same bucket is being emptied at a rate given by $E(t)=\frac{\ln(t+4)}{t+2}$. My ...
3
votes
3answers
78 views

Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$

I was given a task to prove that inequality is true for x>0: $(e^x-1)\ln(1+x) > x^2$. I've tried to use derivatives to show that the $f(x) = (e^x-1)\ln(1+x)-x^2$ is greater than zero, but has never ...
3
votes
3answers
41 views

Trying to show that $\ln(x) = \lim_{n\to\infty} n(x^{1/n} -1)$

How do I show that $\ln(x) = \lim_{n\to\infty} n (x^{1/n} - 1)$? I ran into this identity on this stackoverflow question. I haven't been able to find any proof online and my efforts to get from $\ln(...
1
vote
1answer
151 views

Forward Algorithm Hidden Markov Model matrix help [Discrete]!

So this may seem like a bioinformatics question but it is the math part that is giving me trouble. I'm using a Python package called YAHMM to model DNA sequences. I created a model with two states (...
0
votes
1answer
12 views

Formula for calculating markup with big % for small amounts and small % for larger amounts

I am trying to come up with a formula for calculating markup for products that range in value from a few cents up to tens of Dollars. At 10c I would like the markup to be around 500%, and from 2 ...
4
votes
3answers
151 views

Prove that $f(ab) = f(a) + f(b)$

Question : Assume only that $f: (0,\infty)\to{\mathbb{R}}$ is differentiable and that $f'(x) = 1/x$, and $f(1)=0$. Prove that for all $a,b \in(0,\infty)$, $f(ab)=f(a)+f(b)$. [Hint: Let $g(x)=f(ax)$] ...
5
votes
3answers
256 views

Proving $x$ is a given quotient of logarithms

I'm practicing some questions on logarithms at the moment in order that I'm up to speed with the problem solving aspect before I embark on my PHD in chemical engineering at Boston college next year. ...
1
vote
5answers
50 views

Multiplying two logarithms (Solved)

I was wondering how one would multiply two logarithms together? Say, for example, that I had: $$\log x·\log 2x < 0$$ How would one solve this? And if it weren't possible, what would its domain ...
0
votes
1answer
12 views

Exhaustive search times: 2 to power k = 100 hours - double k, how many hours

An exhaustive search (i.e. checking all combinations of values) takes 100 hours to go through all permutations where a binary key has a length of k. $2^k$ = 100 hours where k is the number of digits ...
0
votes
0answers
46 views

log(x,2log(2x, 2log(2,4x))) >1 , find answers of x

how i slove it , please help me? log(x,2log(2x, 2log(2,4x))) >1 my try: if x>1 =>2log(2x, 2log(2,4x))>x => 2log(2,4x)>(2x)^(x/2) =>4x>2^(((2x)^(x/2))/2) another way log(x,2log(2x, 2log(2,4x)))=...
11
votes
1answer
70 views

If $\log_35=a$ and $\log_54=b$, what is $\log_{60}70$?

One student sent me this question: If $\log_35=a$ and $\log_54=b$, what is $\log_{60}70$? Question asks the value of $\log_{60}70$ in terms of $a$ and $b$. Equations for $a$ and $b$ involved $2$,...
10
votes
5answers
145 views

How do you solve $x^2 = \left(\frac 12\right)^x $?

I'm having trouble finding the steps to solve for $x$. The solutions to this equation are $x=-4$, $x=-2$, and $x=0.76666$ when solved graphically and through the solve function of a TI-nspire cx CAS. ...
11
votes
3answers
486 views

Number system with $e^x = 0$ for some $x$

It is well known that $e^x \ne 0$ for all $x \in \mathbb{R}$ as well as $x \in \mathbb{C}$. Upon reading this article and doing a bit of research I have found that this also applies to the ...
1
vote
2answers
24 views

Cancelling a logarithm

I was wondering if there was a way to cancel out a logarithm? For example: $\log_a A$ > $\log_a B$ What would a have to be for the log to go away and be left with A > B? Thanks in advance!