Question

$\begin{gathered} \green \ast \: { \underline{ \sf{ \pink {solve \: for \: \boldsymbol{r} : }}}} \\ \\ \sf : \implies 4 \boldsymbol{x} = \boldsymbol x \bigg(1 + \dfrac{ \boldsymbol{r}}{100} { \bigg)}^{8} \end{gathered}$
​

Answer 1

Given :-

$\rm :\longmapsto\:4x = x {\bigg(1 + \dfrac{r}{100}\bigg) }^{8}$

To Find :-

• Value of r.

Concept Used :-

Logarithms and Anti-logarithms

Properties of log :-

$\boxed{\bf \: log( {x}^{y} ) = y log(x) }$

$\boxed{\bf \: log(x) = y \implies \: x = {e}^{y} \: or \: x = antilog(y)}$

$\boxed{\bf \: log(2) = 0.3010}$

$\boxed{\bf \: antilog(0.07525) = 1.190}$

Solution :-

Given that

$\rm :\longmapsto\:4 \cancel{x} = \cancel{x} {\bigg(1 + \dfrac{r}{100}\bigg) }^{8}$

$\rm :\longmapsto\:4 = {\bigg(1 + \dfrac{r}{100}\bigg) }^{8}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{ \bf \: Let \:1 + \dfrac{r}{100} = y}$

So,

$\rm :\longmapsto\: {y}^{8} = 4$

Taking log on both sides, we get

$\rm :\longmapsto\: log{y}^{8} = log4$

$\rm :\longmapsto\: 8log{y} = log {2}^{2}$

$\rm :\longmapsto\: 8log{y} = 2log {2}$

$\rm :\longmapsto\: 4log{y} = log {2}$

$\rm :\longmapsto\: 4log{y} = 0.3010$

$\rm :\longmapsto\: log{y} = 0.07525$

$\rm :\longmapsto\: {y} =antilog(0.07525)$

$\rm :\longmapsto\:y = 1.190$

So,

$\rm :\longmapsto\:1 + \dfrac{r}{100} = 1.190$

$\rm :\longmapsto\:\dfrac{r}{100} = 1.190 - 1$

$\rm :\longmapsto\:\dfrac{r}{100} = 0.190$

$\rm :\longmapsto\:r = 100 \times 0.190$

$\bf\implies \:r \: = \: 19$

Additional Information :-

$\boxed{\bf \: log(xy) = log(x) + log(y)}$

$\boxed{\bf \: log(1) = 0}$

$\boxed{\bf \: log(10) = 1}$

$\boxed{\bf \: log( \frac{x}{y} ) = log(x) - log(y)}$

$\boxed{\bf \: log_{x}(y) = \dfrac{ log(y) }{ log(x)}}$

$\boxed{\bf \: log_{ {a}^{x} }( {a}^{y} ) = \frac{y}{x}}$

$\boxed{\bf \: log_{x}(x) = 1}$

Answer 2

Answer:

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Step-by-step explanation:

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