Question

solve for x log x upon log 4 =log 64 upon log 16​

Answer 1

Given that,

$\dfrac{ \log(x)}{ \log(4)} = \dfrac{ \log(64)}{ \log(16)}$

We know that,

• 4 = 2 * 2 = (2²)
• 16 = 2 * 2 * 2 * 2 = (2⁴)
• 64 = 2 * 2 * 2 * 2 * 2 * 2 = (2⁶)

$\longrightarrow \dfrac{ \log(x)}{ \log( {2}^{2} )} = \dfrac{ \log( {2}^{6} )}{ \log( {2}^{4} )}$

We know that,

• $\red{\bigstar} \boxed{\log(a^b) = b \log(a)}$

$\longrightarrow \dfrac{ \log(x)}{ 2\log( {2} )} = \dfrac{ 6\log( {2} )}{ 4\log( {2} )}$

$\longrightarrow \dfrac{ \log(x)}{ 2\log( {2} )} = \dfrac{ 6 \: \: \cancel{\log( {2} )}}{ 4 \: \: \cancel{\log( {2} )}}$

$\longrightarrow \dfrac{ \log(x)}{ 2\log( {2} )} = \dfrac{6}{4}$

$\longrightarrow \dfrac{ \log(x)}{ 2\log( {2} )} = \dfrac{3}{2}$

$\longrightarrow \dfrac{ \log(x)}{ \cancel{2} \: \: \log( {2} )} = \dfrac{3}{ \cancel{2}}$

$\longrightarrow \log(x) = 3 \log(2)$

We know that,

• $\red{\bigstar} \boxed{\log(a^b) = b \log(a)}$

$\longrightarrow \log(x) = \log( {2}^{3} )$

$\longrightarrow \log(x) = \log( 8 )$

Comparing both sides,

$\longrightarrow \underline{ \underline{x = 8 }}$