Questions related to real and complex logarithms.

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1
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2answers
60 views

What is the difference in this question between $\log$ and $\lg$?

Am I right in assuming that $\lg$ just refers to $\log$ base ($10$)? Whereas $\log$ is just any unspecified log? I'm solving $\lg{15}-\lg{5}$ Am I good to just use the standard rules of logarithms, ...
12
votes
3answers
852 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 ...
4
votes
3answers
140 views

$2^n=n$ and similar equations

Is it possible to solve equations in the form $k^n=n$ for n and if so, How? I am new to logarithms and so would be glad if someone could explain even if there is an obvious answer. Also What about ...
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
96 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 $x>2$.
1
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1answer
49 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 $a\gt 0, b\gt 0$: ...
-1
votes
5answers
60 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 $$ ...
-2
votes
1answer
35 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
32 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
17 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
43 views

Logarithm Rules Ambiguity

I'm having some problems explaining myself the following ambiguity. According to logarithm rules: $\ln6=\ln(2\cdot3)=\color\red{\ln2+\ln3}$ ...
0
votes
1answer
44 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
6answers
62 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 ...
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) ...
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
1answer
28 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
23 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
39 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 ...
1
vote
1answer
19 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
33 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 ...
0
votes
1answer
38 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 ...
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 = ...
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
28 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 = ...
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
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
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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 ...
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 ...
3
votes
3answers
33 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 ...
4
votes
3answers
148 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 $g(x)=f(ax)$] ...
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 ...
5
votes
3answers
255 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
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 ...
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
41 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, ...
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 ...
1
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5answers
49 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 ...
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!
1
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2answers
41 views

So many logs with different bases

$ \large { 6 }^{ \log _{ 5 }{ x } }\log _{ 3 }( { x }^{ 5 } ) -{ 5 }^{ \log _{ 6 }{ 6x } }\log _{ 3 }{ \frac { x }{ 3 } } ={ 6 }^{ \log _{ 5 }{ 5x } }-{ 5 }^{ \log _{ 6 }{ x } }$ The sum of ...
0
votes
1answer
19 views

Find unknown x coordinate from log graph

I am not sure where to start on this question. I am not sure how to fit the coordinates into the equation $y=\log_3(x-4)$
10
votes
5answers
141 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. ...
1
vote
1answer
40 views

How to prove derivative of logarithm with base $b$?

I learned how to derive a logarithm with any base. This is the formula: $$\frac{d}{dx}\log_bx=\frac{1}{x\ln b}$$ How can it be proved?
3
votes
5answers
186 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 ...
2
votes
1answer
92 views

How $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots =\ln 2$? [duplicate]

while doing the Integration problem using Limit of a sum approach i have a doubt how $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots =\ln2$$ by infinite geometric series we have ...
5
votes
3answers
494 views

How much proof is needed in such paper (Maths related)?

I'm writing a paper (report) regarding Euler's Number $\space e \space$ (even though he didn't discover it). Within this paper, I show that: $${d\over dx} {e^x} = {e^x}$$ **NOTE: ** This is not ...
0
votes
1answer
87 views

Logarithm in the exponent

$$(2x)^{\log 2} = (3y)^{\log 3} \\ 3^{\log x} = 2^{\log y}$$ Solve for $x$ and $y$. My intuition for solving such problems is taking the logarithm on both sides but it does not work. I also ...
0
votes
1answer
28 views

What is the value of $x$ in this logarithmic inequality? [closed]

Please help me with this inequality : $$\log_2 (x^2-2x) - 3 >0 $$
0
votes
2answers
70 views

If the integral of $c/x$ is $c.log(x)+C$ what is the base?

This question is a follow up to an answer I gave here: How to integrate $1/x$? After the algebra I said that 'This step of course gives the argument of $ln()$ the value $e$ and note that so far we ...