Question

Can anyone help in this questions…………………………………,.,,………………

Answer 1

Solution :-

To Determine: The value of x.

Given Equation :-

$\rm \longrightarrow \ln(x + 3) - \ln(x) = 2 \ln(2)$

We know that :-

$\rm \longrightarrow \ln(x) - \ln(y) = \ln \bigg( \dfrac{x}{y} \bigg)$

Therefore, we get :-

$\rm \longrightarrow\ln \bigg( \dfrac{x + 3}{x} \bigg) = \ln({2}^{2})$

Removing log from both sides, we get :-

$\rm \longrightarrow \dfrac{x + 3}{x} =4$

$\rm \longrightarrow 4x = x + 3$

$\rm \longrightarrow 3x = 3$

$\rm \longrightarrow x = 1$

★ Therefore, the value of x satisfying the given equation is 1.

To Know More :-

Logarithm Formulae.

$\rm 1. \: \: {a}^{n} = b \implies log_{a}(b) = n$

$\rm 2. \: \: log_{a}(1) = 0, \: a \neq0,1$

$\rm 3. \: \: log_{a}(a) = 1, \: a \neq0,1$

$\rm 4. \: \: log_{a}(x) = log_{a}(y) \implies x = y$

$\rm 5. \: \: log_{e}(x) = ln(x)$

$\rm6. \: \: log_{a}(x) + log_{a}(y) = log_{a}(xy)$

$\rm7. \: \: log_{a}(x) - log_{a}(y) = log_{a} \bigg( \dfrac{x}{y} \bigg)$

$\rm 8. \: \: log_{a}( {x}^{n} ) = n\log_{a}(x)$

$\rm 9. \: \: log_{a}(m) = \dfrac{ log_{b}(m) }{ log_{b}(a) },m > 0,b > 0,a \ne1,b \ne1$

$\rm 10. \: \: log_{a}(b) = \dfrac{1}{ log_{b}(a) }$

Answer 2

Answer:

x=1

In is logarithm in mathematics:-

Refer to 1st attachment for working.

Refer to 2nd attachment for identifies.