Question

Solve the log equation for x :-
$x^{\left(\dfrac{\log_3 x + 5}{3}\right)}=10^{\Big(5+\log_{10}x\Big)}}$

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Answer 1

Appropriate Question is

Solve for x :-

$\rm :\longmapsto\:x^{\left(\dfrac{\log_{10} x + 5}{3}\right)}=10^{\Big(5+\log_{10}x\Big)}$

$\large\underline{\sf{Solution-}}$

The given equation is

$\rm :\longmapsto\:x^{\left(\dfrac{\log_{10} x + 5}{3}\right)}=10^{\Big(5+\log_{10}x\Big)}$

Taking log to the base 10 on both sides, we get

$\rm :\longmapsto\: log_{10}\bigg[ x^{\left(\dfrac{\log_{10} x + 5}{3}\right)}\bigg]=log_{10}\bigg[10^{\Big(5+\log_{10}x\Big)}\bigg]$

We know,

$\boxed{ \rm \: log {x}^{y} = y \: logx}$

$\rm :\longmapsto\:\dfrac{1}{3}\bigg[ log_{10}(x) + 5\bigg] log_{10}(x) = \bigg[ log_{10}(x) + 5\bigg] log_{10}(10)$

We know,

$\boxed{ \rm \:log_{10}(10) = 1}$

$\rm :\longmapsto\:\dfrac{1}{3}\bigg[ log_{10}(x) + 5\bigg] log_{10}(x) = \bigg[ log_{10}(x) + 5\bigg]$

$\rm :\longmapsto\:\dfrac{1}{3}\bigg[ log_{10}(x) + 5\bigg] log_{10}(x) - \bigg[ log_{10}(x) + 5\bigg] = 0$

$\rm :\longmapsto\:\bigg[\dfrac{1}{3} log_{10}(x) - 1\bigg]\bigg[ log_{10}(x) + 5\bigg] = 0$

$\rm :\longmapsto\:\bigg[\dfrac{1}{3} log_{10}(x) - 1\bigg] = 0 \: \: or \: \: \bigg[ log_{10}(x) + 5\bigg] = 0$

$\rm :\longmapsto\:\dfrac{1}{3} log_{10}(x)= 1 \: \: or \: \:log_{10}(x)= - 5$

$\rm :\longmapsto\: log_{10}(x)= 3 \: \: or \: \:log_{10}(x)= - 5$

We know,

$\boxed{ \rm \: log_{a}(b) = c \: \bf\implies \:b = {a}^{c}}$

So, using this

$\rm :\longmapsto\:x = {10}^{3} \: \: \: or \: \: \: x = {10}^{ - 5}$

Verification :-

$\red{\rm :\longmapsto\:When \: x = {10}^{3}}$

So, given equation

$\rm :\longmapsto\: {10}^{ 3{\left(\dfrac{\log_{10} {10}^{3} + 5}{3}\right)}}=10^{\Big(5+\log_{10} {10}^{3} \Big)}$

$\rm :\longmapsto\: {10}^{ 3{\left(\dfrac{3 + 5}{3}\right)}}=10^{\Big(5+3 \Big)}$

$\rm :\longmapsto\: {10}^{ 3{\left(\dfrac{8}{3}\right)}}=10^{\Big(8 \Big)}$

$\bf\implies \: {10}^{8} = {10}^{8}$

Hence, Verified

$\red{\rm :\longmapsto\:When \: x = {10}^{ - 5}}$

So, given equation is

$\rm :\longmapsto\: {10}^{ - 5{\left(\dfrac{\log_{10} {10}^{ - 5} + 5}{3}\right)}}=10^{\Big(5+\log_{10} {10}^{ - 5} \Big)}$

$\rm :\longmapsto\: {10}^{ 3{\left(\dfrac{ - 5 + 5}{3}\right)}}=10^{\Big(5 - 5\Big)}$

$\rm :\longmapsto\: {10}^{ 3{\left(\dfrac{0}{3}\right)}}=10^{\Big(0\Big)}$

$\rm :\longmapsto\: {10}^{0} = {10}^{0}$

Hence, Verified