Dolly Robinson
2021-08-17
Answered

Expand using logarithmic properties. Where possible, evaluate logarithmic expressions.

${\mathrm{log}}_{5}\left(\frac{{x}^{3}\sqrt{y}}{125}\right)$

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Alix Ortiz

Answered 2021-08-18
Author has **109** answers

Step 1

On simplification, we get

${\mathrm{log}}_{5}\left(\frac{{x}^{3}\sqrt{y}}{125}\right)={\mathrm{log}}_{5}\left(125\right)\text{}[\because {\mathrm{log}}_{a}\left(\frac{m}{n}\right)={\mathrm{log}}_{a}\left(m\right)-{\mathrm{log}}_{a}\left(n\right)]$

$={\mathrm{log}}_{5}\left({x}^{3}\right)+{\mathrm{log}}_{5}\left(\sqrt{y}\right)-{\mathrm{log}}_{5}\left({5}^{3}\right)\text{}[\because {\mathrm{log}}_{a}\left(mn\right)={\mathrm{log}}_{a}\left(m\right)+{\mathrm{log}}_{a}\left(n\right)]$

$={\mathrm{log}}_{5}\left({x}^{3}\right)+{\mathrm{log}}_{5}\left({y}^{\frac{1}{2}}\right)-{\mathrm{log}}_{5}\left({5}^{3}\right)$

$=3{\mathrm{log}}_{5}\left(x\right)+\frac{1}{2}{\mathrm{log}}_{5}\left(y\right)-3{\mathrm{log}}_{5}\left(5\right)\text{}[\because {\mathrm{log}}_{a}\left({m}^{n}\right)=n{\mathrm{log}}_{a}\left(m\right)]$

$=3{\mathrm{log}}_{5}\left(x\right)+\frac{1}{2}{\mathrm{log}}_{5}\left(y\right)-3\left(1\right)\text{}[\because {\mathrm{log}}_{a}\left(a\right)=1]$

$=3{\mathrm{log}}_{5}\left(x\right)+\frac{1}{2}{\mathrm{log}}_{5}\left(y\right)-3$

On simplification, we get

asked 2022-02-14

Let T be a linear transformation from $R}^{2}\text{}\text{into}\text{}{R}^{2$ such that $T(1,0)=(1,1)\text{}\text{and}\text{}T(0,1)=(-1,1)$ . Find $T(6,1)\text{}\text{and}\text{}T(-5,1)$

$T(6,1)=?$

$T(-5,1)=?$

asked 2020-12-13

To state:

The solution of the given initial value problem using the method of Laplace transforms.

Given:

The initial value problem is,

$y+2{y}^{\prime}+5y=0,y\left(0\right)=2,{y}^{\prime}\left(0\right)=4$

The solution of the given initial value problem using the method of Laplace transforms.

Given:

The initial value problem is,

asked 2022-02-13

Properties of dual linear transformation

Let V,W finite dimensional vector spaces over the field F, and let$T:V\to W$ a linear transformation.

Define$T\cdot :W\cdot \to V\cdot$ the dual linear transformation. i.e $T\cdot \left(\psi \right)=\psi \circ T$ .

Let V,W finite dimensional vector spaces over the field F, and let

Define

asked 2020-12-01

a) To find:

The images of the following points under under a

I.

II.

III, (m,n) interms of m and n

b)To show:

That under a half-turn with the origin as center, the image of a point

c) To find:

The image of

asked 2020-11-23

To find: The solution of the given initial value problem.

asked 2022-02-15

Creating a transformation matrix

This question is related to this one, but more specific. I was given the following question:

Let$D=\{{d}_{1},{d}_{2}\}\text{}\text{and}\text{}B=\{{b}_{1},{b}_{2}\}$ be bases for vector spaces V and W respectively. Let $T:V\to W$ be a linear transformation with the property that $T\left({d}_{1}\right)=-5{b}_{1}-7{b}_{2}\text{}\text{and}\text{}T\left({d}_{2}\right)=9{b}_{1}+4{b}_{2}$ . Find the matrix for T relative to D and B.

This is a pretty simple question and I just took the transformation properties and made a matrix:

$(\begin{array}{cc}-5& -7\\ 9& 4\end{array})$

The answer key has the same answer but with the rows and columns switched:

$(\begin{array}{cc}-5& 9\\ -7& 4\end{array})$

Are these both correct?

This question is related to this one, but more specific. I was given the following question:

Let

This is a pretty simple question and I just took the transformation properties and made a matrix:

The answer key has the same answer but with the rows and columns switched:

Are these both correct?

asked 2021-01-17

Guided Proof Let ${v}_{1},{v}_{2},....{V}_{n}$ be a basis for a vector space V.

Prove that if a linear transformation$T:V\to V$ satisfies

$T({v}_{i})=0\text{}for\text{}i=1,2,...,n,$ then T is the zero transformation.

To prove that T is the zero transformation, you need to show that$T(v)=0$ for every vector v in V.

(i) Let v be the arbitrary vector in V such that$v={c}_{1}{v}_{1}+{c}_{2}{v}_{2}+\cdots +{c}_{n}{V}_{n}$

(ii) Use the definition and properties of linear transformations to rewrite T(v) as a linear combination of$T({v}_{j})$ .

(iii) Use the fact that$T({v}_{j})=0$

to conclude that$T(v)=0,$ making T the zero transformation.

Prove that if a linear transformation

To prove that T is the zero transformation, you need to show that

(i) Let v be the arbitrary vector in V such that

(ii) Use the definition and properties of linear transformations to rewrite T(v) as a linear combination of

(iii) Use the fact that

to conclude that