xlog4 m+1/2 log 4 n-log4 p

2022-04-25
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xlog4 m+1/2 log 4 n-log4 p

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nick1337

Answered 2022-07-19
Author has **564** answers

$x\mathrm{log}\left(4m\right)+\frac{1}{2}\mathrm{log}\left(4n\right)-\mathrm{log}\left(4p\right)$

Simplify each term.

Simplify $\frac{1}{2}\mathrm{log}\left(4n\right)$ by moving $\frac{1}{2}$ inside the logarithm.

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left({\left(4n\right)}^{\frac{1}{2}}\right)-\mathrm{log}\left(4p\right)$

Apply the product rule to $4n$.

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left({4}^{\frac{1}{2}}{n}^{\frac{1}{2}}\right)-\mathrm{log}\left(4p\right)$

Rewrite $4$ as $2}^{2$.

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left({\left({2}^{2}\right)}^{\frac{1}{2}}{n}^{\frac{1}{2}}\right)-\mathrm{log}\left(4p\right)$

Apply the power rule and multiply exponents, $\left({a}^{m}\right)}^{n}={a}^{mn$.

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left({2}^{2\left(\frac{1}{2}\right)}{n}^{\frac{1}{2}}\right)-\mathrm{log}\left(4p\right)$

Cancel the common factor of $2$.

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left({2}^{1}{n}^{\frac{1}{2}}\right)-\mathrm{log}\left(4p\right)$

Evaluate the exponent.

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left(2{n}^{\frac{1}{2}}\right)-\mathrm{log}\left(4p\right)$

Use the quotient property of logarithms, ${\mathrm{log}}_{b}\left(x\right)-{\mathrm{log}}_{b}\left(y\right)={\mathrm{log}}_{b}\left(\frac{x}{y}\right)$.

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left(\frac{2{n}^{\frac{1}{2}}}{4p}\right)$

Simplify each term.

Factor $2$ out of $2{n}^{\frac{1}{2}}$.

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left(\frac{2\left({n}^{\frac{1}{2}}\right)}{4p}\right)$

Cancel the common factors.

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left(\frac{{n}^{\frac{1}{2}}}{2p}\right)$

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$\frac{d{y}_{i}}{dx}={c}_{i}({y}_{i-1}^{3}-{y}_{i}^{3})+{f}_{i}\phantom{\rule{2em}{0ex}}(\text{for}i=1,\dots ,n)$

using the discretised equations, which are presented below in matrix notation:

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Note ${f}_{i}=1$ when $i=1$, but otherwise equals 0 (this considers the initial condition). Whilst I am familiar with Euler's method, I am unsure of the steps to get to the first equation to the second. Can anyone assist me with this? Understanding this is necessary for a project I'm working on, so I would really appreciate any help.

$\frac{d{y}_{i}}{dx}={c}_{i}({y}_{i-1}^{3}-{y}_{i}^{3})+{f}_{i}\phantom{\rule{2em}{0ex}}(\text{for}i=1,\dots ,n)$

using the discretised equations, which are presented below in matrix notation:

$\mathit{y}({t}_{k+1})={\mathit{C}}_{1}\mathit{y}({t}_{k})+{\mathit{C}}_{2}{\mathit{y}}^{3}({t}_{k})+\mathit{f}$

Note ${f}_{i}=1$ when $i=1$, but otherwise equals 0 (this considers the initial condition). Whilst I am familiar with Euler's method, I am unsure of the steps to get to the first equation to the second. Can anyone assist me with this? Understanding this is necessary for a project I'm working on, so I would really appreciate any help.

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Where c is the given curve

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