Question

if 3 power x =5 power x-2

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Answer 1

Correct question :-

If 3ˣ = 5ˣ⁻², then find the value of x.

Answer :-

Value of x is 2log 5/(log 5/3)

Solution :-

3ˣ = 5ˣ⁻²

Taking log on both sides

⇒ log 3ˣ = log 5ˣ⁻²

⇒ xlog 3 = (x - 2)log 5

[ ∵ log aᵐ = mlog a]

⇒ xlog 3 = xlog 5 - 2log5

⇒ 2log5 = xlog 5 - xlog 3

⇒ 2log 5 = x(log 5 - log 3)

⇒ 2log 5 = x(log 5/3)

[∵log x - log y = log x/y]

⇒ 2log 5/(log 5/3) = x

⇒ x = 2log 5/(log 5/3)

∴ the value of x is 2log 5/(log 5/3)

Answer 2

Given

${3}^{x} = {5}^{(x - 2)}$

To find

The value of x

Consider

${3}^{x} = {5}^{(x - 2)}$

Applying log on both sides

$log {3}^{x} = log {5}^{(x - 2)}$

we know that

$\underline{\sf loga^m=mloga}$

Using the formula

$xlog3 = (x - 2)log5$

$= > xlog3 = xlog5 - 2log5$

$= > xlog3 - xlog5 = - 2log5$

$= > x(log3 - log5) = - 2log5$

$= > x = \dfrac{-2log5}{log3 - log5} \\ = > x = \frac{2log5}{log5 - log3}$

Using the below formula

$\underline{\sf logm-logn=log\dfrac{m}{n}}$

$\boxed{\sf x=\dfrac{2log5}{log\dfrac{5}{3}}}$

Some more important formulas:

$= > log1 = 0$

$= > logm + logn = logmn$

$= > logm - logn = log \dfrac{m}{n}$

$= > log {a}^{n} = nloga$