Question

Q.3]Find the value of x if 'the Xth power of 4 is 6

Answer 1

Answer:

The value of x is 1.292 approximately.

Step-by-step-explanation:

We have given that,

4ˣ = 6

We have to find the value of x.

Now,

$\displaystyle{\sf\:4^x\:=\:6}$

Taking log to the base 4 on the both sides, we get,

$\displaystyle{\sf\:\log_4\:(\:4^x\:)\:=\:\log_4\:(\:6\:)}$

$\displaystyle{\implies\sf\:x\:\times\:1\:=\:\log_4\:(\:6\:)\:\qquad\cdots[\:\log_b\:(\:b^k\:)\:=\:k\:]}$

$\displaystyle{\implies\sf\:x\:=\:\log_4\:(\:6\:)}$

We know that,

$\displaystyle{\pink{\sf\:\log_b\:(\:a\:)\:=\:\dfrac{\log_x\:(\:a\:)}{\log_x\:(\:b\:)}}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{\log_2\:(\:6\:)}{\log_2\:(\:4\:)}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{\log_2\:(\:6\:)}{2}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{1}{2}\:\times\:\log_2\:(\:6\:)}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{1}{2}\:\times\:\log_2\:\left(\:4\:\times\:\dfrac{3}{2}\:\right)}$

We know that,

$\displaystyle{\pink{\sf\:\log_b\:(\:xy\:)\:=\:\log_b\:(\:x\:) \:+\:\log_b\:(\:y\:)}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{1}{2}\:\times\:\left[\:\log_2\:(\:4\:)\:+\:\log_2\:\left(\:\dfrac{3}{2}\:\right)\:\right]}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{1}{2}\:\times\:\left[\:2\:+\:\log_2\:\left(\:\dfrac{3}{2}\:\right)\:\right]}$

We know that,

$\displaystyle{\pink{\sf\:\log_b\:\left(\:\dfrac{x}{y}\:\right)\:=\:\log_b\:(\:x\:)\:-\:\log_b\:(\:y\:)}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{1}{2}\:\times\:[\:2\:+\:\log_2\:(\:3\:)\:-\:\log_2\:(\:2\:)\:]}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{1}{2}\:\times\:[\:2\:+\:\log_2\:(\:3\:)\:-\:1\:]}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{1}{2}\:\times\:[\:2\:-\:1\:+\:\log_2\:(\:3\:)\:]}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{1}{2}\:\times\:[\:1\:+\:\log_2\:(\:3\:)\:]}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{1\:+\:\log_2\:(\:3\:)}{2}}$

$\displaystyle{\implies\sf\:x\:\approx\:\dfrac{1\:+\:1.584}{2}}$

$\displaystyle{\implies\sf\:x\:\approx\:\dfrac{2.584}{2}}$

$\displaystyle{\implies\underline{\boxed{\red{\sf\:x\:\approx\:1.292}}}}$