Solve the given problem giving your answer as a single logarithm

[chwala]

Homework Statement

This is a past paper question. I have concern with the term single logarithm. Kindly see attached.

Relevant Equations

logs

Markscheme;

Not many international students would understand single logarithm as expected by examiners. In my take $x = \dfrac{\ln 2}{2}$ is single logarithm and therefore a full $4$ marks ought to be awarded.

In any case, the form;

$\dfrac{\ln 2}{2} = \ln 2^\frac{1}{2}$

are one and the same.

Kindly clarify on this.

 

[scottdave]

I guess they want everything "inside 1 logarithm"

 

[Mark44]

In any case, the form; $\dfrac{\ln 2}{2} = \ln 2^\frac{1}{2}$

I guess they want everything "inside 1 logarithm"

Or stated equivalently, as the logarithm of a single expression, such as $\ln 2^\frac{1}{2}$.

 

[chwala]

Or stated equivalently, as the logarithm of a single expression, such as $\ln 2^\frac{1}{2}$.

So which is which... Student to lose final mark because of the requirement to express solution as per examiners perspective?

 

[WWGD]

Homework Statement: This is a past paper question. I have concern with the term single logarithm. Kindly see attached.
Relevant Equations: logs

View attachment 346419

Markscheme;

Not many international students would understand single logarithm as expected by examiners. In my take $x = \dfrac{\ln 2}{2}$ is single logarithm and therefore a full $5$ marks ought to be awarded.

In any case, the form;

$\dfrac{\ln 2}{2} = \ln 2^\frac{1}{2}$

are one and the same.

Kindly clarify on this.

View attachment 346420

Where does the $k$ in $lnk$ come from ( 3rd line from top, left side)?

 

[chwala]

Where does the $k$ in $lnk$ come from ( 3rd line from top, left side)?

I do not understand your phrase. Did you understand my concern?

 

[WWGD]

I do not understand your phrase. Did you understand my concern?

This comes from $ln(a^b)=bln(a)$. Is that what you were asking?
I was referring to the 3rd cell down from the top , that makes reference to $k$, without specifying what it is.

 

[chwala]

The mark scheme is attached sir, the full mark is allocated only if solution is expressed as indicated something that I have a problem with ... The previous steps drawing 3 marks also ought to have drawn the full 4 marks.

 

[Mark44]

Where does the $k$ in $lnk$ come from ( 3rd line from top, left side)?

What you're seeing as 'k' is actually 'x'.

 

[Mark44]

So which is which... Student to lose final mark because of the requirement to express solution as per examiners perspective?

From the mark scheme it seems pretty clear to me that full credit would be given for either $\ln(\sqrt 2)$ or $\ln(2^{1/2})$ (i.e., as the logarithm of a single expression) but possibly some deduction for $\frac{\ln 2}2$.

 

[WWGD]

From the mark scheme it seems pretty clear to me that full credit would be given for either $\ln(\sqrt 2)$ or $\ln(2^{1/2})$ (i.e., as the logarithm of a single expression) but possibly some deduction for $\frac{\ln 2}2$.

The mark scheme or the Mark44 scheme?

 

[chwala]

What you're seeing as 'k' is actually 'x'.

I think the $k$ refers to the number $2$ (a constant, that is) and not $x$.

 

[SammyS]

I think the $k$ refers to the number $2$ (a constant, that is) and not $x$.

Right you are.

Here are some alternative forms of the solution as a single logarithm.

Note that if $\displaystyle \ \{ A,\,B \} \ge 1\, ,\$ then $\displaystyle \log_A B = \dfrac{1}{\log_B A}\ .$

So, $\displaystyle \ x = \dfrac{\ln 2}{2}=\dfrac{\log_{\Large{e}} 2}{2}=\dfrac{1}{2\, \ \log_2 e}=\dfrac{1}{ \ \log_2 e^2}\ .$

As well: $\displaystyle \ x =\log_{\Large{(e^2)}} 2 \ .$