Question

# Solve the equations and inequalities. Write the solution sets to the inequalities in interval notation. log2(3x−1)=log2(x+1)+3

Logarithms

Solve the equations and inequalities. Write the solution sets to the inequalities in interval notation. $$\displaystyle{\log_{2}}{\left({3}{x}−{1}\right)}={\log_{2}}{\left({x}+{1}\right)}+{3}$$

2021-02-19

Property used:

Property 1:

$$\log_{2}(m)-\log_{2}(n)=\log_{2}(\frac{m}{n})$$

Now to simplifying the given equation:

$$\log_{2}(3x-1)=\log_{2}(x+1)+3$$

$$\log_{2}(3x-1)-\log_{2}(x+1)=3$$

$$\log_{2}(\frac{3x-1}{x+1})=3$$ [Using Property 1.]

Now taking antilog 2 and solving:

$$(\frac{3x-1}{x+1})=2^{3}$$

$$(3x-1)=8(x+1)$$

$$(3x-1)=8x+8$$

$$3x-8x=8x+1$$

$$-5x=9$$

$$x=-\frac{9}{5}$$

Since,

The solution does not satisfy the given equation.

Hence there is no solution for $$x \in R$$