Question

if {2}^{x + 1} = {3}^{1 - x} then find the value of x.​

Answer 1

The required value of x is given by :

$x = \frac{\ln 1.5}{\ln 6}$

Step-by-step explanation:

$\begin{gathered}2^{x+1}=3^{1-x}\\\\\text{Taking natural log on both the sides}\\\\\implies (x+1)\ln 2=(1-x)\ln 3\\\\\implies x\ln 2+ \ln 2=\ln 3-x\ln 3\\\\\implies x\ln 2+x\ln 3=\ln 3-\ln 2\\\\\implies x(\ln 2 +\ln 3) = \ln 3-\ln 2\\\\\implies x\ln 6=\ln 1.5\\\\\implies x = \frac{\ln 1.5}{\ln 6}\end{gathered}$

Hence, The required value of x is given by :

$x = \frac{\ln 1.5}{\ln 6}$

Answer 2

Solution!!

→ 2ˣ⁺¹ = 3¹⁻ˣ

Take the logarithm of both sides of the equation.

→ x + 1 = log₂ (3) - log₂ (3)x

Move the expression to the left-hand side and change its sign.

→ x + 1 + log₂ (3)x = log₂ (3)

Move the constant to the right-hand side and change its sign.

→ x + log₂ (3)x = log₂ (3) - 1

Factor out x from the expressions.

→ (1 + log₂ (3))x = log₂ (3) - 1

Divide both sides of the equation by 1 + log₂ (3).

→ x = (log₂ (3) - 1)/(1 + log₂ (3))

1 can be expressed as a logarithm with the same base and argument.

→ x = (log₂ (3) - 1)/(log₂ (2) + log₂ (3))

Use logₐ (x) - logₐ (y) = logₐ (x/y) to simplify the expression.

→ x = (log₂ (3/2))/(log₂ (2) + log₂ (3))

Calculate the sum.

→ x = (log₂ (3/2))/(log₂ (6))

Use (logₙ (x))/(logₙ (a)) = logₐ (x) to simplify the expression.

→ x = log₆ (3/2)