Find the value of ##N## in the logarithm problem

[chwala]

Homework Statement

If $2log_8N=p$, $log_2 (2N)=q$, $q-p=4$ then find $N$

Relevant Equations

Logs

My approach is as follows;
$$\log_8 N= \frac {1}{2} p$$
$$\log_2 (2N)=q$$
$$→8^{\scriptstyle\frac 1 2} = N$$
$$ 2^q=2N$$
$$2^{\scriptstyle\frac 3 2} =N$$
$$2^q= 2N$$
then from 1 and 2, it follows that,
$$2^{q-1.5p} =2,$$ on solving the simultaneous equation;

$$q-1.5p=1, q-p=4$$, we get $q=10$ and $p=6$
$2^{10}=2N$
$N=512$

any other ways...cheers guys

 

[pasmith]

$$\begin{split} 4 &= \log_2(2N) - 2\log_8(N) \\ &= 1 + \log_2(N)\left(1 - \frac{2}{\log_2(8)} \right) \\ &= 1 + \log_2(N)\left(1 - \frac{2}{3}\right)\\ &= 1 + \tfrac{1}{3}\log_2(N)\end{split}$$ Hence $N = 2^{9}$.

 

[BvU]

Homework Statement:: If $2log_8N=p$, $log_2 2N=q$, $q-p=4$ then find $N$
Relevant Equations:: Logs

My approach is as follows;
$$log_8 N= \frac {1}{2} P$$
$$log_2 2N=q$$
$$→8^{0.5p} = N$$
$$ 2^q=2N$$
$$2^{1.5p} =N...1$$
$$2^q= 2N...2$$
then from 1 and 2, it follows that,
$$2^{q-1.5p} =2,$$ on solving the simultaneous equation;

$$q-1.5p=1, q-p=4$$, we get $q=10$ and $p=6$
$2^{10}=2N$
$N=512$

any other ways...cheers guys

Hello there !

Not much room for improvement, but I'm prepared to play devil's advocate

Clarity:

Don't write $\log_2 2N=q\qquad$ but $\log_2( 2N)=q$​

Don't use $\rightarrow\$ but $\Rightarrow\$ (##\Rightarrow##, $\$not ##rightarrow##)​

(single arrow means 'goes to', double arrow means if 'left' is true then 'right' is true )​

​

Consistency:

Don't write $\log_8 N= \frac {1}{2} P\qquad$ but $\log_8 N= \frac {1}{2} p$​

(case consistency is important)​

$\LaTeX$:

Use $\log\$, $\$ not $log\$ (##\log##, $\$not ##log##)​

(log is an operator name, not the product of the three variables $l$, $o$ and $g$)​

(See 'Special functions' here)​

​

Use $\frac 1 2 \$ and $\frac 3 2\$ , not $\scriptstyle {0.5} \$ and $\scriptstyle {1.5} \$.​

​

Often $\frac 1 2 \$ (\scriptstyle \frac 1 2 when in displayed math mode) looks better than $\displaystyle \frac 1 2 \$. And vice versa (\displaystyle \frac 1 2 when in in-line $\TeX$ mode).​

​

Displayed math allows you to combine a few things on a line:​

$$\qquad \log_8 N= {\scriptstyle \frac {1}{2}} p\qquad \Rightarrow\qquad 8^{0.5p} = N\qquad \Rightarrow\qquad 2^{\frac 3 2 p}=N$$ $$\log_2( 2N)=q \qquad \Rightarrow\qquad 2^q= 2N$$

Dont use $...1\$ etc for equation numbers.​

Alignment is a nightmare and it looks ugly. Instead, learn about environments​

(See 'Multiple lines' here. To suppress equation numbers confusion, use {align*} and manually add equation numbers with ##\tag 1## etc. )​

​

My 'best shot' (still unhappy because the double arrows don't align )​

$$\begin{align*}
\log_8 N&= {\scriptstyle \frac {1}{2}} p & \Rightarrow \qquad 8^{\frac 1 2p} &= N &\Rightarrow 2^{\frac 3 2 p}&=N\tag 1 \\
\log_2( 2N)&=\ q & \Rightarrow \qquad 2^q&= 2N \tag 2 \end{align*} $$

​

$\$​

 

[chwala]

Hello there !

Not much room for improvement, but I'm prepared to play devil's advocate

Clarity:

Don't write $\log_2 2N=q\qquad$ but $\log_2( 2N)=q$​

Don't use $\rightarrow\$ but $\Rightarrow\$ (##\Rightarrow##, $\$not ##rightarrow##)​

(single arrow means 'goes to', double arrow means if 'left' is true then 'right' is true )​

​

Consistency:

Don't write $\log_8 N= \frac {1}{2} P\qquad$ but $\log_8 N= \frac {1}{2} p$​

(case consistency is important)​

$\LaTeX$:

Use $\log\$, $\$ not $log\$ (##\log##, $\$not ##log##)​

(log is an operator name, not the product of the three variables $l$, $o$ and $g$)​

(See 'Special functions' here)​

​

Use $\frac 1 2 \$ and $\frac 3 2\$ , not $\scriptstyle {0.5} \$ and $\scriptstyle {1.5} \$.​

​

Often $\frac 1 2 \$ (\scriptstyle \frac 1 2 when in displayed math mode) looks better than $\displaystyle \frac 1 2 \$. And vice versa (\displaystyle \frac 1 2 when in in-line $\TeX$ mode).​

​

Displayed math allows you to combine a few things on a line:​

$$\qquad \log_8 N= {\scriptstyle \frac {1}{2}} p\qquad \Rightarrow\qquad 8^{0.5p} = N\qquad \Rightarrow\qquad 2^{\frac 3 2 p}=N$$ $$\log_2( 2N)=q \qquad \Rightarrow\qquad 2^q= 2N$$

Dont use $...1\$ etc for equation numbers.​

Alignment is a nightmare and it looks ugly. Instead, learn about environments​

(See 'Multiple lines' here. To suppress equation numbers confusion, use {align*} and manually add equation numbers with ##\tag 1## etc. )​

​

My 'best shot' (still unhappy because the double arrows don't align )​

$$\begin{align*}
\log_8 N&= {\scriptstyle \frac {1}{2}} p & \Rightarrow \qquad 8^{\frac 1 2p} &= N &\Rightarrow 2^{\frac 3 2 p}&=N\tag 1 \\
\log_2( 2N)&=\ q & \Rightarrow \qquad 2^q&= 2N \tag 2 \end{align*} $$

​

$\$​

Thanks bvu, I am learning this beautiful latex language. Cheers mate.