Questions related to real and complex logarithms.

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0answers
71 views

Maclaurin Expansion of $\ln(3+x)$

I'm currently evaluating a simple Maclaurin expansion, the confusion I have with is why the expansion of this function is constructed to be: $\ln\left[3\left(1+\dfrac{x}{3}\right)\right]$ as opposed ...
6
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4answers
171 views

What are some methods to show $\log$ is not a rational function?

It is easy to show $\log$ isn't a polynomial (no continuous extension to $\mathbb{R}$). More challenging is showing it isn't rational. Suppose it were a rational function. Then write, the fraction ...
3
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2answers
48 views

How do I simplify this Log with a Fraction in it?

So I have: $$ \log_2(5x) + \log_2 3 + \frac{\log_2 10}{2} $$ I understand that when there is addition, and the bases are the same, I can simply multiply what is in the parenthesis. So for the first ...
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1answer
34 views

Find how many years must elapse before the proportions of red kangaroos and grey kangaroos are reversed, assuming the same rates continue to apply.

I have this question (sorry I'm not able to embed it): Q.7. There are approximately ten times as many red kangaroos as grey kangaroos in a certain area. If the population of grey kangaroos ...
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1answer
35 views

Codomain of p-adic logarithm

We have the natural map $$\log: \mathbb{C}^\times \to \mathbb{R}$$ $$z \to \log |z|$$ Is there a p-adic analogue of this? By this I mean, a map $\log_p: \mathbb{C}_p^\times \to \mathbb{Q}_p$, ...
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1answer
101 views

It's correct to say that $\log _{ 1 }{ 1 } =1\quad $ & $0$? [duplicate]

$\log _{ 1 }{ 1 } =1\quad $ v $0$? Because $1^1 = 1$ and $1^0 = 1$?
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2answers
46 views

Solving for x with logarithms

I've been asked to solve for $x\,$ in $5^x + 4·5^{x+1} = 63$ The answer is $x = \frac{\log3}{\log5}$ I cannot do this without a calculator. Is there a particular method I should be using ...
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1answer
167 views

How to prove $f(x) = \ln x$ continuous by proving first that $f(x)$ continuous at $1$, and then by using $\ln (xy) = \ln(x) + \ln(y)$. [duplicate]

I have a question concerning the proof of the continuity of $f(x) = \ln x$. I read in a comment by Pedro Tamaroff to ncmathsadist's answer to this question that this can be proved in two steps: ...
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2answers
61 views

How to solve for log with a number outside?

$$\log_6(4x-10)+1 = \log_6(15x+15)$$ This is a sample problem. I know that when the bases of log are the same, all you have to do is set the parenthesis inside equal to each other. If the $1$ wasn't ...
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1answer
27 views

Summation Closed form for floor$\left(\log_n\right)$

The closed sum for the floors of logs of consecutive integers is: $$\sum_{i=0}^n \lfloor \log_2i\rfloor = n\lfloor \log_2n\rfloor-2^{\lfloor \log_2n\rfloor+1}+\lfloor \log_2n\rfloor+2$$ This formula ...
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1answer
24 views

Principal argument summation

Let $\text{Arg}$ be a an principal argument in $(-\pi, \pi]$. I know that, for all $z_1,z_2\in\mathbb{C}\setminus \{0\}$, the expression $\text{Arg}(z_1z_2)= \text{Arg} z_1 + \text{Arg} z_2$ doesn't ...
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2answers
34 views

Log equation $\log(2x-1) = -x+3$ with two non log values [closed]

What is the correct approach to solving a log equation with more than one non log value? Please demonstrate using the following equation: $$\log(2x-1)=-x+3$$
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0answers
28 views

Why is $\log z = \ln r + i\theta$ ($r>0, \alpha <\theta < \alpha + 2\pi$) discontinuous at $\alpha$?

In one book on complex variables it is written that, given the function $\log z = \ln r + i\theta$ (for proper citation, let's call it function (2), as in the book) ($r>0, \alpha <\theta < \...
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3answers
49 views

Prove $\sum_{k = 2}^\infty \ln(1+\frac{1}{k^2})$ converges using $\exp(x) \geq 1+x$.

All I've got so far is $$\exp(x) \geq 1+x \Rightarrow x \geq \ln(1+x) \Rightarrow \frac{1}{k^2} \geq \ln\left(1+\frac{1}{k^2}\right)$$ which (since $\ln(1+\frac{1}{k^2})$ is larger than zero) means ...
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4answers
58 views

What does $e^{a*ln(x)}$ equal in terms of $a$ and $x$, and how is this found?

I saw somewhere that it would be $x^a$, but I'm not sure why.
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1answer
36 views

Find $f(r)$ for which $\int \log(x^2 + y^2) f(x^2 + y^2) dx$ has simple form

Consider the following antiderivative $$ F(x, y) = \int \log (x^2 + y^2) f(x^2 + y^2) dx. $$ I'm looking for some function $f(r)$ with following properties: $f(r)$ is uniformly bounded $f(r)$ "...
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1answer
45 views

Solving $\log(x-2) + \log(9-x) \lt 1$.

The solution. Now, in comment section, a person has mentioned (and it's given in the answer behind the book) that another solution could be $2 \lt x \lt 4$. I've tried numerous times, but have not ...
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3answers
98 views

Ambiguous conditions for $x$ in logarithm?

I have just realised that there is a little ambiguity in defining conditions for $x$ in logarithm. Let me illustrate it on a simple example: $\log{x}$ is valid for $x>0$ , $2\log{x}$ is also ...
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7answers
126 views

Finding $\lim_{x\to 0} \frac{\ln(x+4)-\ln(4)}{x}$ without L'hôpital

I have this $\lim_{}$. $$\lim_{x\to 0} \frac{\ln(x+4)-\ln(4)}{x}$$ Indetermation: $$\lim_{x\to 0} \frac{\ln(0+4)-\ln(4)}{0}$$ $$\lim_{x\to 0} \frac{0}{0}$$ Then i started solving it: $$\lim_{x\to 0} \...
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1answer
16 views

A number based on which the logarithmic function outputs negative values for proper fractions only?

Let this number be $B$, $B$ achieves the following: $\frac{\ln(x)}{\ln(B)}>0$, for x>1 $\frac{\ln(x)}{\ln(B)}<0$, for 1>x>0 If such a number is worth serious attention, what is its name by ...
3
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0answers
55 views

All solutions of $ z^i = i^z $

In the simple equation $ z^i = i^z $ how are all complex values found? $ z= \pm \, i, $ and what else? It can be found by inspection, but to find general solution: We take logs, there is a ...
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1answer
23 views

Logarithm common factor problem

there exist positive A, B, and C, with no common factor greater than 1, such that $A.log_{200}5 +B.log_{200}2=C$ what is A + B +C I dont know how to equal this equation
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2answers
39 views

Solving a logarithmic equation where the logarhitm is exponentiated

I have troubles solving the following logarthitmic equation. $$ \ 2(\log_x{\sqrt7})^2-\log_x{\sqrt7}-1 =0 $$ The results are supposed to be $ \ x_1 = {\frac{1}{7}}, x_2 = \sqrt7 $ I have tried ...
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1answer
27 views

What is the difference between a Logarithm and Scientific Notation? [closed]

Why would you use Logarithms over Scientific Notation and vice versa, since they generally serve some of the same functions?
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2answers
33 views

Subtracting Multiple Logarithms

I am asked to simplify $$J=\ln (x^2-16)-\ln x-\ln(x-4), \quad x>4$$ Since each logarithm's argument is non-negative, I can use $$\ln x -\ln y=\ln\frac{x}{y}$$ and obtain the correct answer $$\ln\...
3
votes
1answer
106 views

What should $\int \frac{1}{x} dx$ equal to?

Before you say that $\int \frac{1}{x} dx$ is equal to $\ln|x| +C$ due to positve and negative, I would like to show you why it is not convincing to me. Problem 1 and its possible solution. \begin{...
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4answers
92 views

I'm stuck in a logarithm question: $4^{y+3x} = 64$ and $\log_x(x+12)- 3 \log_x4= -1$

If $4^{y+3x} = 64$ and $\log_x(x+12)- 3 \log_x4= -1$ so $x + 2y= ?$ I've tried this far, and I'm stuck $$\begin{align}4^{y+3x}&= 64 \\ 4^{y+3x} &= 4^3 \\ y+3x &= 3 \end{align}$$ $$\begin{...
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5answers
416 views

Can someone help me with this question of finding x as exponent?

The equation is: $$6^{x+1} - 6^x = 3^{x+4} - 3^x$$ I need to find x. I forgot how to use logarithm laws. Help would be appreciated. Thanks.
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0answers
24 views

$\log(x)$ as iteration-series: how can this be made correct?

I was tinkering with the question whether the logarithm $\log(x)$ can be expressed by some more useful series than by the Mercator series (in terms of (1+x)) for a certain question. One idea ...
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2answers
89 views

Solving $y^y = x$ for large $x$

I was playing around with recurrence relations and noticed that $\sqrt x$ has the fun property that $$\frac{x}{f(x)} = f(x)$$ ($\sqrt{x}$ and its negation are the only functions $f(x)$ that satisfy ...
0
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1answer
26 views

Exponent to maximize the expression $log_b \left(a\frac{b-1}{b^k-1}\right)$

Given $ a, b \in \mathbb N $, how to maximize the expression $$ log_b \left(a\frac{b-1}{b^k-1}\right) \in \mathbb N $$ Put differently, what is the minimum $k \in \mathbb N $ verifying $$ a\frac{b-1}{...
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1answer
53 views

How to solve this equation using logarithms?

I have to solve for all real values of $x$. $(5+2\sqrt6)^{x^2-3}+(5-2\sqrt6)^{x^2-3}=10$ I tried to take $\log_{10}$ on both sides but could not do this. How do I do this?Thanks for any hint or ...
0
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2answers
41 views

Minimum of the antilogarithm

Given a $ a \in \mathbb N $, what is the lowest $ b \in \{1, ..., a \} $ for which $ log_b a \in \mathbb N $ ? How to compute this function in a non-iterative way? Examples (even if too obvious): $ a ...
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3answers
51 views

Proof of a logarithm identity

I would like to know how to prove the following log identity: $x^{\frac{\log(\log(x))}{\log(x)}} = \log(x)$
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2answers
79 views

Proving that two branch cuts can cancel out

Define the following functions $\mathbb{C}\to\mathbb{C}:$ $$u(z)=\frac{\log \left(z+\frac{1}{2}\right)}{z}\quad \left[-\pi\leqslant\arg \left(z+\tfrac12\right)<\pi\right];\quad v(z)=\frac{\log z}{z}...
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1answer
26 views

During the 56th month or the 57th month?

A car depreciates in value according to the model $$V=Ak^t$$ where £$V$ is the value of the car $t$ months from when it was new. Its value when new was £$12499$ and $36$ months later its value was £$...
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2answers
50 views

Complex logarithms when computing real-valued integral

My question arise when I try to calculate real-valued integral, specifically, I want to evaluate the integral \begin{equation} \int_0^1 \frac{\ln \left(\frac{x^2}{2}-x+1\right)}{x} dx \end{equation} ...
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7answers
1k views

What is “8 log 2”? [closed]

When someone says "8 Log 2" what does this equate to in writing? Does it mean the following? $$ \log _{2} 8 $$ And if so, what is the value of this? When I plug those numbers into this log ...
2
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3answers
118 views

How to solve $3(a+1)(b+1)=3^a \times 2^b$?

Hi I'm new to logarithms and not sure how to solve equations involving logarithms. I managed to find this equation to answer a problem solving question, however now I do not know how to solve the ...
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2answers
32 views

Graphing log with number in front of “log”

When I have something like $y = log_2(x)$ I know that I have to turn it into exponential form and get: $2^y = x$. Next, I make a table for $X,Y$ and choose about 5 values for $y$, typically $-1, 0, 1, ...
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0answers
33 views

Order Size estimation of converging sum used for approximation of logarithm

I know it can be shown that $\log n=\sum_{i=1}^\infty \frac{(n-1)^i}{in^i}$ for $\forall n\in\Re$ where $n\ge1$ For given natural m, I tried to find the order size of k = f(m,n) in order for the ...
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0answers
22 views

Proof this limit superior is finite.

Let $\{ w_n \}$ be a sequence of non-negative numbers and put $M_n=\sum_{k=1}^n w_k^2 \xrightarrow{n\to\infty} \infty $. Proof that $$\limsup_{n\to\infty} \dfrac{\ln \ln \sqrt{M_n \ln \ln M_n} }{\ln \...
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2answers
34 views

Logs - changing the base to evaluate

Just a bit confused about how to evaluate the following $$\log_3 8\times \log_5 9\times \log_2 5$$ What I have done so far: I have used the change of base rule to change each log to base $3$, ...
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2answers
34 views

log to exponential form, but with number in front of log

So I understand how to put a log equation into exponential form. For example, $y = \log_2(x)$ is $2^y = x.$ However, I don't understand what to do when there is a number in front of $\log$, such as $...
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7answers
94 views

Solve the following equation : $\log_2(x)*\log_4(x)*\log_8(x)=4.5$

I have the following equation : $$\log_2(x)*\log_4(x)*\log_8(x)=4.5$$ Usually, I do post what I made to do, but in this case a friend of mine tackle me with this question after I didn't mess with ...
3
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2answers
80 views

Is the value of $\log_27$ a rational number?

Is $\log_27$ a rational number?
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4answers
67 views

Use the definition of $\ln(x)$ as an integral to show that $f(x)=\frac{ln(x)}{x^2} \leq 1/x$ for all $x\geq 1$.

As the title says, if we let $$f(x)=\frac{ln(x)}{x^2}$$ I know that $$ln(x)=\int_1^x \frac{dt}{t} dt$$ Since $x^2 >0$ we can rewrite the question as $ln(x) \lt x$ $\forall$ $x\ge1$ How do we ...
1
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1answer
32 views

Logarithmic Taylor series question [closed]

Consider the transformation of variables, x = $\frac{y+1}{y-1}$ How would you develop log(x) as a Taylor Series in y about zero?
0
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2answers
20 views

How to show that: $\log_a (x^{a}-x)-\log_a \Big(\dfrac{x^{a}-x}{a}\Big)=1$, where $a$ and $x$ are positive integers.

I was studying Fermat's Little Theorem and Logarithm to see if there is any interesting result or correlation exist between the two. So I came up with this equation. I know few basic logarithmic ...
2
votes
2answers
52 views

$\ln(x)$ and Big O notation

I have tried to assert that $\ln(x)=O(x^0)$ a few times, but it seems fairly obvious that this statement should be false, and so I've been faced with some rightful speculation. My reason is that $$\...