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Can someone explain why $x^{\log (a)} = a^{\log (x)}$ ?

Danika Mckay 2022-10-24

re-arrange equation $L = 2^{10 (v - 1)} v^{2}$
Is it possible to re-arrange this equation to make v the subject?
$L = v^{2} . 2^{10 (v - 1)}$
If so, what is the answer?
If it helps (which by excluding zero it should)...
$0 < v < 1$
I have tried pages of scribbling and got nowhere. In desperation I have tried solving each product separately (easy enough), and then tried to get the right overall result from some combination of $\sqrt{L}$ and $(l o g 2 (L) + 10) / 10$ but that hasn't got me anywhere.
I am out of my depth. Please help.

Ralzereep9h 2022-10-24

Finding out the logarithmic function for the situation below
The situation reads as follows:
There are 3000 barbs in a pond and every year 20% of the barbs die and then 1000 new barbs come to the pond. A logarithmic function needs to be plotted to graph this change in population.
I worked through a part of the above situation and arrived at the function:
$y = 3000 ({0.8}^{x}) + 1000 ({0.8}^{x - 1}) + {0.8}^{x - 2} + \dots + {0.8}^{1} + {0.8}^{0}$
How do I convert this above equation into a logarithmic function?

gasavasiv 2022-10-24

Solving a logarithmic equation with variables on each side
Okay, so while doing a problem for my calculus class I was required to graph two functions in order to see where they intersect, as according to my teacher there is no way to solve it analytically.
This really bothers me and there must be a way to solve it. I did research online to try to solve it, but I had no idea where to even start.
Here is the equation.
$y = \ln (5 - y)$

Amina Richards 2022-10-24

Integration - involving logarithm
I have a slight confusion on how to integrate functions of the form:
$\int \frac{a}{x} d x$
Suppose we have the following function:
$\int \frac{- 2}{x} d x$
There are two ways we can proceed to integrate this function. One is to treat the $- 2$ sign as a constant and take it out of the integration function:
$- 2 \int \frac{1}{x} = - 2 \ln x = \ln \frac{1}{x^{2}}$
Another way to do this is to treat the $- 2$ as part of the integration variable:
$\int \frac{1}{- 0.5 x} d x = - 2 \ln (- 0.5 x)$
This seems to obtain two different answers. Which method is correct? Or is there something that I'm missing?

Pellagra3d 2022-10-24

Find the derivative of the following function: $y (x) = (\ln (2 x))^{5 x}, x > \frac{1}{2}$

Jack Ingram 2022-10-24

Integration problem: $\int \ln (\sin (\sqrt{x}) + \cos (\sqrt{x})) d x$
I really don't know how to start solving this problem; any tips or solutions will be greatly appreciated.
Thank you in advance.

Trace Glass 2022-10-24

Is there any expansion for $\log (1 + x)$ when $x > 1$ ?
Is there any expansion for $\log (1 + x)$ when $x > 1$ ?

Josiah Owens 2022-10-23

Solve for $x, y$ : $(3 x)^{\log 3} = (4 y)^{\log 4}$ and $4^{\log x} = 3^{\log y}$
then how do I solve for x?
I tried taking log on both sides but after few steps I got stuck

Wyatt Weeks 2022-10-23

If the positive numbers x,y,z are in harmonic progression, then $\log (x + z) + \log (x - 2 y + z)$ equals
$a) 4 \log (x - z) b) 3 \log (x - z) c) 2 \log (x - z) d) \log (x - z)$
How do i approach this problem? IF x,y,z are in HP, $\Rightarrow y = 2 x y / x + z$

bergvolk0k 2022-10-23

Find all values of $x$
$\log_{2} (2^{x - 1} + 3^{x + 1}) = 2 x - \log_{2} (3^{x})$
Let $u = 2^{x}$ and let $y = 3^{x}$
For ease, let $\log_{2}$ be represented by just $\log$ so:
Then, $\log (u / 2 + 3 y) = \log (u^{2}) - \log (y)$ , which means, $\log (u / 2 + 3 y) = \log (u^{2} / y)$ and so:
$u / 2 + 3 y = u^{2} / y ⟹ \frac{u y}{2} + 3 y^{2} - u^{2} = 0$
Factoring a little, $y (u / 4 + 3 y) + u (y / 4 - u) = 0$ Doesn't help, replace back:
$2^{x - 1} 3^{x} + 3^{2 x + 1} - 2^{2 x} = 0$ . But this is all I can go upto? A hint?

Sonia Elliott 2022-10-23

Is the complex square root of $z^{2} = \pm z$ ?
Is $\sqrt{z^{2}} = \pm z$ , for $z$ complex?
I think it is, since either $- z$ or $+ z$ satisfies the definition
$\sqrt{z^{2}} = e^{\frac{1}{2} \log (z)^{2}}$
but I just wanted to make sure. It's a bit tricky going from the positive square to the complex square root.
Thanks,

Aydin Jarvis 2022-10-23

Solving $(\log_{2} x) - 4 = 5 \log_{x} 2$
How do I solve the following equation?
$(\log_{2} x) - 4 = 5 \log_{x} 2$
I tried doing this:
log (x * X/32) = 4 2 and i got x^2 = 512
Also, how do I solve equations of the form log⋅log⋅=… ? logarithms

Antwan Perez 2022-10-23

Exponents with the same power
I've wanted to practice solving simple operations on exponents, so I've made a couple of equations to which I know the answers.
$5^{x} - 4^{x} = 9$
I feel really stupid, because I can't solve this one (obviously the answer is 2). I can't separate the x's after using logarithms.
Any hints? Thanks.

Emmy Swanson 2022-10-23

Finding the leading exponent of a binary number
Let's say that the binary representation of a number $k$ is $2^{X_{n}} + 2^{X_{n - 1}} + \dots + 2^{X_{0}}$ with each term in this polynomial having a $1$ or $0$ multiplied to it.
Now, given only $k$ , and no other information, is there a way that I can find the number $X_{n}$ (which is the highest degree of the binary representation of $k$ )?

raapjeqp 2022-10-23

How to solve the following equation? $\log_{3} (\log_{x} (\log_{4} 16)) = - 1$
I am trying to solve this equation for $x$ . This is what I have so far:
$\log_{3} (\log_{x} 2) = - 1.$
Okay, now I have this:
$\log 2 = (1 / 3) \log x$
How do I isolate x from here?

Madilyn Quinn 2022-10-23

Showing an inequality with ln
I have to show that the following inequation is true:
$\frac{\ln (x) + \ln (y)}{2} \leq \ln (\frac{x + y}{2})$
I transformed it into
$\frac{\ln (x \cdot y)}{2} \leq \ln (x + y) - \ln (2)$
because I thought that I better can show the inequation here, but I don't know how to proceed.
How can I proceed or am I completely wrong?

Cale Terrell 2022-10-22

Limits involving logarithm and argument in the complex plane

$Log ((2 / n) + 2 i)$ as $n \to \infty$
$Log (2 + (2 i / n))$ as $n \to \infty$
$Arg ((1 + i) / n)$ as $n \to \infty$
$(Arg (1 + i)) / (n)$ as $n \to \infty$

For the Log questions, I am getting $(i π) / 2 + \log (2)$ for the first problem, then for the second I am getting only $\log (2)$ . Because the Log's in the questions are capitalized, I think I may have to add on $2 π i k$ to each of the answers. Is that correct?
for the last two problems (the Arg problems) I got zero for both because as n goes to infinity the n is the denominator for each so I thought they probably each go to zero. but also, the Arg is capitalized here as well, so I am getting the feeling I am doing these wrong. Can anybody help? Thanks!

Brianna Schmidt 2022-10-22

Principal square root of a product of complex numbers with positive real part
Given n complex numbers $z_{i}$ with $ℜ z_{i} > 0$ , why is it that
$\prod_{i} \sqrt{z_{i}} = \sqrt{\prod_{i} z_{i}} ?$
Numerically, this appears to be the case, however, I don't see an easy way to prove it.

Cyrus Travis 2022-10-22

Proof the expession $\log_{12} 18 \times \log_{24} 54 + 5 (\log_{12} 18 - \log_{24} 54) = 1$
I am trying to proof the following expression (without a calculator of course).
$\log_{12} 18 \times \log_{24} 54 + 5 (\log_{12} 18 - \log_{24} 54) = 1$
I know this isn't a difficult task but it's just killing me. I have tried many things, among which was base transformation to 12 and expressing every logarithm in terms of $\log_{12} 3$ and $\log_{12} 2$ but every time I try to do it, I mess up something. I don't know if my concentration is terrible or I'm doing something wrong.
Thanks ;) ( if there are more levels to this task, I'd like a hint, not a complete solution)

Finding logarithms questions and answers online is always pure luck, especially if these are provided with detailed answers to questions. This is exactly what you can find below regardless of your practical application, college course, or complexity. These logarithms questions are related to higher education courses, which is why they will help you find the answers to what you currently have. When you deal with logarithmic equations you are approaching an exponential equation where the variable appears in an exponent as it’s in (log b (bx) = x. It also shows that the logarithmic graph equation can be represented as well.

Algebra II

Learn How to Graph Logarithmic Equations