Questions related to real and complex logarithms.
6
votes
4answers
209 views
Solving the exponential equation: $3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0$
I have this exponential equation that I don't know how to solve:
$3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0$ with $x \in \mathbb{R}$
I tried to factor out a term, but it does not help. ...
13
votes
3answers
368 views
Are the logarithms in number theory natural?
I find the frequent emergence of logarithms and even nested logarithms in number theory, especially the prime number counting business, somewhat unsettling. What is the reason for them?
Has it maybe ...
0
votes
1answer
48 views
Formula to calculate when you will have a certain amount of money in your bank account
What is the formula to calculate when I will have a million dollars in my bank account?
An example is that I have $\$6,000$ in my account and have $6\%$ interest rate on that.
How long will it take ...
4
votes
2answers
225 views
Can all logarithm problems be solved algebraically?
Trying to solve $\log_2(x-1)=\log_3(x+1)$ and can't seem to get it algebraically. Tried changing bases, moving things around, but can't seem to crack it.
4
votes
1answer
216 views
The definition of the logarithm.
One usually gets several definitions of the logarithm along his studies.
You might be first introduced to the exponential and then told that the logarithm is its inverse.
You might be given
$$\log ...
1
vote
3answers
1k views
From natural log to log base 10
The constraints of this question is related to a programming problem, but I must get the math right in order for it to be applied to code. The actual problem is I need a function that evaluates to log ...
5
votes
2answers
243 views
continuum between linear and logarithmic
A friend and I are working on a heatmap representing some population numbers. Initially we used a linear color scale by default. Then, because the numbers covered a wide range, I tried using a log ...
6
votes
2answers
178 views
Prove $ n+1<\frac{\log 4}{\log3}+\frac{\log 44}{\log33}+\frac{\log4444}{\log3333}+\cdots+\frac{\log 444\ldots444}{\log333\ldots333} <n+2 $
Prove that
$$ n+1<\frac{\log 4}{\log3}+\frac{\log 44}{\log33}+\frac{\log4444}{\log3333}+\frac{\log 44444444}{\log33333333}+\cdots+\frac{\log 444\ldots444}{\log333\ldots333} <n+2$$
where last ...
4
votes
1answer
75 views
Does the logistic function really uniquely satisfy this?
It is said that the logistic function (denoted $y(u)$ below) is derived from the relation:
$$\frac{dy}{du}=y(u)(1-y(u))$$
Does $y(u)=\frac{1}{1+e^{-u}}$ really uniquely satisfy this? I don't see ...
2
votes
1answer
274 views
closed form solution for summation of $\log(i)$
Is there a way to find a closed form solution for:
(Note that base is $2$)
$\displaystyle\sum_{i=1}^n\log_2(i)$
thanks for any help
Can't find a formula for this
1
vote
2answers
163 views
Graph of a Log Function
I am curious as to why Wolfram|Alpha is graphing a logarithm the way that it is. I was always taught that a graph of a basic logarithm function $\log{x}$ should look like this:
However, ...
0
votes
1answer
223 views
Reverse of a log-based f(x)?
I am attempting to create a function, f(y) that can reverse my existing function, f(x).
...
2
votes
1answer
152 views
Are all logarithms multiple of each other?
I was doing a time complexity problem, and the solution mentioned that there is a single class for logs. Ie. we can write $\log_a (x) = \Theta(\log_b(x))$ where $a$ is not equal to $b$.
This can be ...
5
votes
1answer
99 views
Solutions for this logarithmic equation.
For which values of $k$ does the equation
$\log_a(kx+3)+\log_a(x+1)=\log_a(2x+1)$
have one or more solutions in $x$?
The logarithmic functions must have the restriction that the argument is ...
2
votes
3answers
198 views
Logarithms of the form $x=e^y$
I have the following math problem:
The number of people in a town of 10,000 who have heard a rumor started by a small group of people is given by the following function: $N(t) = ...
1
vote
3answers
521 views
Solve 10 base logarithms
I'm a n00b in math and I wanted to know how should I solve ten base logarithms. e. g.:
Log 40 to base 10
Thanks in advance.
2
votes
3answers
187 views
Solving $\int\frac{\ln(1+e^x)}{e^x} \space dx$
I'm trying to solve this integral.
$$\int\frac{\ln(1+e^x)}{e^x} \space dx$$
I try to solve it using partial integration twice, but then I get to this point (where $t = e^x$ and $dx = \frac{1}{t} ...
1
vote
3answers
87 views
Comparing numbers in form $x^y$
Let's consider two numbers in form $x_1^{y_1}$ and $x_2^{y_2}$
How can we compare those two numbers without evaluating them ?
Can we use logarithms to check it ? If yes - how ?
Thanks in advance.
...
5
votes
3answers
217 views
Does $\log(ab)^n$ equal $(\log(a)+\log(b))^n$ or $n\log(a)+n\log(b)$?
I think this might be a case of slight ambiguity in notation, but here goes:
On a test question, I was required to expand the expression $\log (ab)^n$. Since the logarithm is a function, I reasoned ...
0
votes
3answers
307 views
arithmetic progression involving logarithm
$\log_2 X$, $\log_2 (X+9)$ and $\log_2(X+45)$ are 3 consecutive terms of an arithmetic progression; find
$\qquad$(i) the value of X;
$\qquad$(ii) the first term and the common difference; and
...
3
votes
2answers
90 views
Limit with prime sequence and inverse logintegral
I found formula below$$\lim_{n\to\infty}\frac{\operatorname{li^{-1}}(n)}{p_n}=1$$ where $\operatorname{li^{-1}}(n)$ is inverse logintegral function and $p_n$ is prime number sequence.
Can anyone ...
5
votes
2answers
205 views
Why isn't $\frac{\mathrm{d} }{\mathrm{d} x} \ln(x)$ specified as $\frac{1}{x},x>0$?
As I understand, $\begin{eqnarray} \frac{\mathrm{d}}{\mathrm{d}x}\ln(x)\end{eqnarray} $ is generally specified as $\begin{eqnarray} \frac{1}{x} \end{eqnarray}$. Would it be more appropriate to state ...
2
votes
3answers
484 views
How to evaluate $\int_{1}^{2}\frac{dx}{1+x+\ln x}$?
Can you help me find the value of the integral
$$\int_{1}^{2}\frac{dx}{1+x+\ln x}$$
Thank you
5
votes
1answer
203 views
Inverse function of $\operatorname{li}(x)$ over $x>\mu$?
How can I get the inverse function of $\operatorname{li}(x)$ over $x>\mu$?
Where $$\operatorname{li}(x)=\int_{0}^{x}\frac{ds}{\ln(s)}$$ is the so-called logarithmic integral, and ...
2
votes
1answer
75 views
How to get “N” from $k=\log_2(N)$
I know it's a easy question but unfortunately I forgot some school stuff:
I have $k=\log_2(N)$ and want to know $N$.
Is it $N=2^k$ while using $2$ as base?
Short comments are welcome :)
1
vote
5answers
392 views
How to solve for negative numbers in logarithmic equations
I am trying to solve the equation
$$z^n = 1.$$
Taking $\log$ on both sides I get $n\log(z) = \log(1) = 0$.
$\implies$ $n = 0$ or $\log(z) = 0$
$\implies$ $n = 0$ or $z = 1$.
But I clearly missed ...
2
votes
1answer
123 views
Find the limit of a sequence defined as solution to equation
We can easily prove that the equation of variable $x$
$$(E_{n}): \frac{x(\ln x)^n}{1+x}=\frac{e}{2(e+1)}$$
has a unique solution $u_{n}$ in $[1,e]$ for all integers $n$ greater than $1$. Let's call it ...
3
votes
2answers
73 views
Trying to figure out how an approximation of a logarithmic equation works
The physics books I'm reading gives $$\triangle\tau=\frac{2}{c}\left(1-\frac{2m}{r_{1}}\right)^{1/2}\left(r_{1}-r_{2}+2m\ln\frac{r_{1}-2m}{r_{2}-2m}\right).$$
We are then told $2m/r$
is small for ...
3
votes
3answers
214 views
Inequality for logarithms
I conjecture the following inequality is true
$$\ln x \le (x - 1)\ln\frac{x}{x-1}$$
for all $x > 1$, but I cannot give a proof.
I will appreciate if someone can provide one.
0
votes
1answer
137 views
How to solve a literal equation
How do I solve $2^{x-1}=3^{x+a}$? I cannot solve it and have spent an hour on it trying many different ways. Please help me! Thank you!
3
votes
1answer
165 views
predicting runtime of $\mathcal{O}(n \log(n))$ algorithm, one “input size to runtime” pair is given
I'm given the runtimes for input size $n=100$ of some polynomial-time (big-Oh) algorithms and an $\mathcal{O}(n \log(n))$ one. I want to calculate the runtimes for:
$200$, $1000$ and $10000$.
For the ...
2
votes
2answers
128 views
How to solve $5 - \log_2 (x - 3) = \log_2(x+1)$
Sorry I have to ask such a simple question, my brain is fried after today.
After substituting with a system of equation, I end up with this "simple" logarithmic problem.
$$5 - \log_2 (x - 3) = ...
2
votes
1answer
202 views
How to find logarithms of negative numbers?
Logarithms of negative numbers must be complex.
But how do you find $\ln{(-2)}$ expressed in something like $x \cdot i$ where $x \in \mathbb{R}$?
3
votes
2answers
98 views
Simple logarithmic equation
If $y=Ae^{-kt}$ and $y=19.6$ when $t=2$, and $y=19.02$ when $t=5$, find the value of the constants $A$ and $k$. Give your answers correct to $2$ decimal places.
I have spent a while (an hour+) on ...
1
vote
1answer
120 views
From half to double, linear to logarithmic scale.
I am making a game where you want a skill value to modify some in game values. With a scale that goes from half to double. 50% to 200%. If I'd do it linear 125% will be the centre but I want the ...
3
votes
3answers
132 views
Squeeze an integral
Would you have any idea about this problem ?
Prove that for all nonnegative integers $n$, the following inequalities hold:
$$\frac{e^2}{n+3}\leq \int_{1}^{e} x (\ln x)^n \,dx \leq ...
0
votes
1answer
113 views
Logarithmic differentiation for this function
What is the value of $f'(x)$ at $c$, when $f(x) = \log_x c = e$?
(I understand the answer could be $1/e$) but am unable to substantiate the reasoning. Can someone please help me take the ...
2
votes
1answer
122 views
Find solution to system with logarithms
I have two equations:
\begin{align*}
3 \ln x + \ln y &= 3 \\
4 \ln x - 6 \ln y &= -7 \\
\end{align*}
Do I just proceed as I have learned with adding equations resulting in:
\begin{align*}
...
1
vote
1answer
70 views
Basic logarithm simplification
From $\displaystyle \frac{\log_n b}{\log_n a}$, how do we get $\log_a b$ using algebra?
I haven't been able to do it for about an hour now; I would love some help! Thanks!
2
votes
2answers
2k views
The difference between log and ln
$$\dfrac{1}{2}\ln(x+7)-(2 \ln x+3 \ln y)$$
Our professor let's us solve this but i do not understand how $\ln$ works. He says it has same properties with $\log$ but i still don't get it. What's the ...
1
vote
3answers
169 views
question about differentiation on logarithmatic functions
I am a little confused as to why sometimes they will treat $x$ in $\ln x$ as $\ln ab$ and other times its treated as $\ln x$.... To be more clear, here is an example:
$2\ln(3x^2-1)$ and $\ln2x$. ...
1
vote
1answer
380 views
Compute lambda of Poisson distribution
$p = \lambda e^{ -\lambda }$
If $p$ and $e$ are known. How can I calculate $\lambda$?
I tried on log on both side but it did not help. Any suggestions?
0
votes
1answer
44 views
Solve in terms of $b$: $\log_b (1 - 3x) = 3 + \log_b x$
$$\log_b (1 - 3x) = 3 + \log_b x$$
If I use the properties of logs, I end up with
$$\log_b \left(\frac{1 - 3x}{x}\right) = 3$$
From there, the example I have says to exponentiate both sides, ...
2
votes
3answers
118 views
Solving a logarithmic equation
$\ln(x + 1) = 2 + \ln(x - 1)$; solve for $x$.
From there I get
$$\ln \frac{x+1}{x-1} = 2.$$
Am I headed in the right direction, in our examples we would exponentiate both sides, does that still ...
0
votes
2answers
122 views
Any idea how to solve this equation?
Any idea how to solve this equation?
$$x^2\log_{3}x^2-(2x^2+3)\log_{9}(2x+3)=3\log_{3}\frac{x}{2x+3}$$
0
votes
2answers
99 views
Calculating dB output from this example
This is my extra credit assignment so don't tell me answers, but please guide me how I should do this. I need to learn.
Question states: Determine the power output of the receiver in watts and in the ...
3
votes
3answers
59 views
Determine if the equation is valid/true
The equation is:
$$\log_b \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}} = 2\log_b(\sqrt{3}+\sqrt{2}).$$
I can get as far as:
$$\log_b(\sqrt{3}+\sqrt{2}) - \log_b(\sqrt{3}-\sqrt{2}) = ...
0
votes
2answers
306 views
Express the logarithm in terms a,b,c
Suppose that:
$\log_{10}A = a$
$\log_{10}B = b$
$\log_{10}C = c$
I need to express the following in terms of $a$,$b$,$c$.
$\log_{10}A + 2\log_{10}(1/A)$
$\log_{10}(((AB)^5)/C)$
...
2
votes
2answers
65 views
Evaluate expression using change of base
This is an awkward question to me, it was not covered in class.
Suppose:
$$\begin{align*}
\log_b 2 &= A, \\
\log_b 3 &= B, \\
\log_b 5 &= C.
\end{align*}
$$
Then, use the ...
1
vote
2answers
73 views
Logarithms equation, litteral
I have problems with the following logarimthic equation:
$$\log _a \left(\frac{x+\sqrt{x^2+5}}{5}\right) = b$$
How can I compute $ \log _a (x-\sqrt{x^2-5})$ in terms of $b$?
