Questions related to real and complex logarithms.

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If $a²+b²=7ab$ where a and b are positive then show that $log(1/3(a+b))=1/2(log a +log b)$

Welcome sir, to the content of my question, please help me: If $a²+b²=7ab$ where a and b are positive then show that $log(1/3(a+b))=1/2(log a +log b)$ Thank you sir.
3
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3answers
109 views

Regarding Cauchy - Goursat theorem with Log function

If $f(z) = \operatorname{Log}{z+2} $ and $C$ is $|z|=1$ , can Cauchy-Goursat theorem be applied at all? I was having the impression that log function resemble a ray $$\ln{r}+i\theta$$ therefore there ...
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1answer
45 views

Calculating an exponentially increasing vector of points in a test and measure system

My application is setting and measuring current and voltage in a physical system with a software algorithm. Given these parameters: min, ...
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1answer
39 views

Find the $\log_y(x)$ given $x$ and $y$ in the form of a number.

So I was doing logs trying to reteach myself pre-calculus and I noticed I don't remember how to turn logs in to numbers. I tried to calculate the $\log_48$ so I turned it in to $4^x=8$ and then ...
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1answer
31 views

how to graph logarithmic and exponential equations

I am not looking for the answer to the question, it would be helpful but an explanation would also be very helpful. How would you graph the following equations? ln is log base e, while log is log ...
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1answer
35 views

tricky logarithm substitutions, precalculus

For all three, the second part in the ellipses is the question. For example, the first one would be log base B (M) = X ... then ... log base M (B^2) = ? http://i.imgur.com/J2gdQbS.gif ...
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2answers
68 views

Solving a logarithmic equation. I need help on solving them when they are in exponents.

$$x^{\large 2\log^3x-3/2\log x}=\sqrt{10}$$ Can someone help me to solve it? Also, when we have $2\log^2 x$, is it equal to $4(\log x)^2$
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3answers
117 views

It's a logarithmic worksheet and O can't solve it.

$\log_ax=p$, $\log_bx=q$ , $\log_{abc}x$=r. What is $\log_cx$?.. It's on my math homework can someone solve it cause I need it.
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2answers
247 views

What do logarithms distribute over?

I notice that division distributes over addition Root extraction distributes over multiplication What operator do logarithms distribute over: ie: what non-constant function $H \in C^2 \rightarrow C$ ...
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2answers
102 views

Intuition behind log plotting

The two plots are the same, except the 2nd one has been log transformed on the y axis. Could I please draw your attention to what happens when Theta >0.65? After taking the log-lin plot it seems ...
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6answers
248 views

Prove that for all $x>0$, $1+2\ln x\leq x^2$

Prove that for all $x>0$, $$1+2\ln x\leq x^2$$ How can one prove that?
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1answer
509 views

convexity of log of moment generating function

Why is log of a moment generating function of random variable Z is convex? that is $\log \mathbb{E}[\exp(\lambda.Z)]$ My logic says since expectation is linear so it is in particular convex and ...
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1answer
47 views

Maclaurin series of $\ln(2+x^2)$

Find the Maclaurin series of $\ln(2+x^2)$. I know that $\displaystyle\ln(1+x) = \sum_{n=1}^\infty\frac {(-1)^{n-1}} {n} x^n $ So is $\displaystyle\ln(1+x^2) = \sum_{n=1}^\infty \frac ...
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2answers
73 views

Logarithm Question $\log_{3}9x+\log_{3}x=4$

Edit: Mistake was at $\log_{3}9x^2=4$ solution shown below Confused on this question, not sure what I did wrong here. $$\boxed{\log_{3}9x+\log_{3}x=4}$$ $$\log_{3}9x+\log_{3}x=4$$ $$\log_{3}9x\cdot ...
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1answer
157 views

Complex Analysis: Log Function

I want to approach this problem with maximum understanding of everything that is going on. I have the function $F(z)=\log(z^2+4)$, and I want to give a region in which it is analytic. I guess I ...
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4answers
189 views

How do you solve this exponential equation?

$3(16)^x+2(81)^x=5(36)^x$ How do you change the bases to combine the terms? The correct answer should be 0 and 0.5. Edit: So this equation can't be solved algebraically? I have to use creative logic ...
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2answers
106 views

How do you solve logarithmic equations like this one?

How do you solve $$3\log(x-15)=\left(\frac{1}{4}\right)^x?$$ The solution is approximately $16$. How would you solve a logarithmic equation with an solution approximately equal to a number without ...
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1answer
116 views

Baby Rudin Excercie 1.7 (Existence of Logarithms)

I'm currently working through Baby Rudin and need help on exercise 1.7(f). The question: Let $A$ be the set of all $w$ such that $b^w < y$, and show that $x = \text{sup}(A)$ satisfies $b^x = ...
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1answer
105 views

How to solve this logarithm without using the change of base formula?

I'm doing an assignment on logarithms, and I've stumbled upon a tricky question. The task looks like this: http://puu.sh/5Gcll.png For the first 3 I have no problem. However, for d) I have no idea ...
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1answer
131 views

How to prove $\operatorname{Log}(z) = \log(|z|)+i\arg(z)$.

The value of the principal branch of the logarithm can be evaluated by the formula \begin{align*} \operatorname{Log}(z) = \log(|z|)+i\arg(z), \end{align*} where $\arg(z) \in (-\pi,\pi)$ and ...
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1answer
40 views

Calculating complex logarithm

I have to calculate the following log: a) log(-4) b) log (3i) I don't really know what to do.. a) $ log(-4) = log|-4| + i\cdot arg(-4) + 2ki\pi = log4 + ?? + 2ki\pi$ b) $ log(3i) = log|3i| + i\dot ...
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1answer
30 views

How to solve the following logarithm?

How to solve this?? The solution is 1001.
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2answers
90 views

Solve $\frac{\log(2x+1)-\log 4}{1-\log(3x+2)}=1$

My attempt:$$\frac{\log(2x+1)-\log4}{1-\log(3x+2)}=1$$ $$\frac{\log(2x+1)-\log4}{\log10-\log(3x+2)}=\log10$$ $$\frac{\log\frac{(2x+1)}{4}}{\log\frac{10}{(3x+2)}}=\log10$$ ...
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4answers
135 views

Solve$(log_{2}(x+1))^2=4$

$$(log_{2}(x+1))^2=4$$ $$log_{2}(x+1)*log_{2}(x+1)=log_{2}16$$ $$x^{2}+2x-15=0$$ $$(x+1)*(x+1)=16$$ $$x^{2}+2x+1=16$$ $$x^{2}+2x-15=0$$ $$(x+5)(x-3)=0$$ $$x_1=-5; x_2=3$$ The solution is only $x_1=3$. ...
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1answer
74 views

simplify the following logarithm and roots expression

please , I need help to solve this problem Can anyone solve it simplify the following $\log{\sqrt[4]{729\sqrt[3]{\dfrac{1}{39}\sqrt{\left(\dfrac{1}{27}\right)^4}}}}$ thank in advance to all
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2answers
38 views

Growth rate of $1/(\log(x)-\log(x-1))$

Let $x>1$ be a real number. Let $y=\dfrac{1}{\log(x)-\log(x-1)}$. My question: Approximately how fast does $y$ grow (asymptotically) in terms of $x$? (e.g. linear, polynomial, exponential)?
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2answers
75 views

Why is $x\log(x)$ convex?

Why is $x\log(x)$ convex? According to the definition it must hold: $(tx+(1-t)y)\log(tx+(1-t)y)\le tx\log(x)+(1-t)y\log(y)$ for all positive $x,y$ and $t\in[0,1]$ edit: It is allowed to ...
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3answers
48 views

logarithms and function

If $\log_{2}(f(x)+|\sin x|)=\log_{2} x$ then: A) $f(x)>0$ for each $x \in R$ B) $\lim_{x\to\infty}f(x)= +\infty$ C) the function is strictly increasing D) $f(\pi)=\pi$ So firstly I define ...
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1answer
107 views

Complex exponent properties?

Here is a line in a proof in a complex analysis text: $\sqrt{1-z^2}=\sqrt{1-z}\sqrt{1+z}$ I know you can't do this in general, but when can you do it? Here is what I tried: ...
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3answers
95 views

need help with solving logarithmic equations

No clue how to approach this problem..
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2answers
716 views

Confusion between negative and positive signs in natural logarithm

If $z= - 0.1887\cdot(x^{0.7637})\cdot(y^{0.2306})$ Its natural logarithm will be $\ln(z) = - [ \ln(0.1887) + 0.7637 \ln(x) + 0.2306 \ln(y)]$ or $\ln(z) = - [ \ln(0.1887) - 0.7637 \ln(x) - 0.2306 ...
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4answers
54 views

Logarithm / exponential equation, not sure what to make of this, (simple)

$$(2 \log_a x)(3 \log_{x^2} 4) = 3$$ No idea how to approach this problem other than moving the 2 and the 3 into an exponent..
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4answers
55 views

How do you solve this logarithm?

Solve for $x$ in the following: $$x = 9^{\log_{3}\left(2\right)}$$ The answer is $4$, but why?
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2answers
688 views

Don't understand simple logarithm problem with fractional base

$$3\log_{\frac{4}{9}}\sqrt[4]{\frac{27}{8}}$$ $$\log_{\frac{3}{2}}\frac{16}{81}$$ I understand using the expansion property to expand the division into a subtraction but how do I proceed from there? ...
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2answers
116 views

Simple Log Question

Fairly new to logs, stuck here: The equation $y=4^{3x}$ can be written in terms of $x$: $$y=4^{3x}$$ $$\log(y)=\log(4^{3x})$$ $$0=3x\log(4)-\log(y)$$ $$0=3x\log\bigg(\dfrac{4}{y}\bigg)$$ At this ...
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3answers
83 views

The inverse function of a logarithm equation

I've tried many things with this question, and just can't seem to get it quite right, can someone please show me how to answer this question? Thank you in advance. $$g(x) = \ln(5x+25) \qquad g^{-1}(x) ...
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1answer
44 views

Problem with re-arranging a solution of mine

So this is the question: $\color{darkblue}{h(x)=4\exp(x-4)}\qquad\qquad h^{-1}(x)=\ln\left(\dfrac{\boxed{\phantom{X}}}{\boxed{\phantom{X}}}\right)\,\boxed{\phantom{XXX}}$ It wants me to enter in ...
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2answers
79 views

Finding the inverse of a natural log

How would I find the inverse of $$\ln(8x-64)?$$ I've tried put $8x-64$ as the power to the base of $e$, I don't know what to do from there on, thanks in advance
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3answers
163 views

Is there a logarithm function through this three given points?

I've got the task to find a logarithm function which contains the following points: $$\begin{align*} A&(5 \mid 4)\\ B&(3\mid6)\\ C&(2\mid8.5) \end{align*} $$ Now I need to find the ...
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1answer
86 views

Is anyway to prove this: $\prod_{k=1}^{n}(a_{k})< (1/n^n)*(\sum_{k=1}^{n}(\sqrt{1+a_{k}*a_{k+1}}))^n$

$$ \prod\limits_{k=1}^{n}a_{k} < {1 \over n^{n}}\left(\,\sum_{k = 1}^{n}\,\sqrt{1+a_{k}\,a_{k+1}\,}\,\right)^n $$ ak and n are positive real number greater than 0. EDIT: a_{k+1} becomes a_{1} ...
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4answers
370 views

Solve $2^{x}=x^{2}$

I've been asked to solve this and I've tried a few things but I have trouble eliminating x. I first tried taking the natural log: $x\ln \left( 2\right) =2\ln \left( x\right) $ $\dfrac {\ln \left( ...
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1answer
227 views

What is natural logarithm of this expression?

What is natural logarithm of this expression? $y = 4*[x^9*x^6]$ Is it $\ln(y) = 4 * [ 9\ln(x) + 6\ln(x)]$ or $\ln(y) = \ln(4) + 9\ln(x) + 6\ln(x)$
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1answer
102 views

Solve the inequality $2^{\left( x^{3}-x\right) } < 1$

$2^{\left( x^{3}-x\right) } < 1$ Let $2^{\left( x^{3}-x\right) }-1=f\left( x\right)$ To find the values for which $f(x)<0$ I let $f(x)=0$: $2^{\left( x^{3}-x\right) }-1=0$ $2^{\left( ...
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1answer
122 views

Precalculus - Exponential and Logarithmic Equations

Mike Kallenberg deposited some money in a bank account that earns 5.6% interest compounded continuously. How long would it take to double the amount in money in Mr. Kallenberg's account?
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2answers
85 views

logarithms equations, different bases

solve equations: $\log_x 10 +2\log_{10x} 10-3\log_{100x} 10=0$ so I tried to use $\log_a b=\frac{1}{\log_b a}$ but it didn't work for me.
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1answer
215 views

Show whether $\log r$ has a conjugate harmonic function on $\mathbb{C} \setminus \{0\}$

Can someone help me understand this passage in a student-written wiki article? The question is whether $u(x+iy) = \log \sqrt{x^2+y^2}$ has a conjugate harmonic function on $\mathbb{C}\setminus \{0\}$. ...
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2answers
3k views

What's the formula to solve summation of logarithms?

this is my first question here. I'm studying summation and everything I know is that: $\sum_{i=1}^n\ k$ is $\frac{n(n+1)}{2}\ $ $\sum_{i=1}^{n}\ k^2$ is $\frac{n(n+1)(2n+1)}{6}\ $ $\sum_{i=1}^{n}\ ...
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0answers
48 views

A problem on logarithm

Can this expression (see below) be written in the form $g_kw^{h_k}$. Where $g_k$ and $h_k$ are functions of only $k$?: $(1- k)^{\lceil\log_kw\rceil - 1}$. Here $k$ and $w$ are positive integers. I ...
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3answers
84 views

Satisfying equality between logarithmic expressions

Apologies in advance for any misused terminology, or if this is the wrong place for the question (I think it's okay though). I am given a group of logarithmic expressions such as: $- (a \log(a) + ...
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2answers
6k views

Writing an equation for a log function given the graph

I have the following graph for a logarithmic function $f$: I don't know any thing about writing an equation for a logarithmic function by knowing it's graph. All what I know is how to draw a graph ...