Question

What is the true solution to the logarithmic equation below? log₂ (6x)-log₂ (√x)=2

Answer 1

Answer:

The true solution is x=4/9

EXPLANATION

The logarithmic equation given to us is

We need to use the quotient rule of logarithms.

When we apply this law the expression becomes

We now take the antilogarithm of both sides to get

We square both sides to get,

We evaluate to obtain,

This simplifies to

We divide both sides by 36 to get

We simplify to get,

Answer 2

answer: x =  $\\ \frac{4}{9\\}$

$log_{2}$ (6x) - $log_{2}$ ($\sqrt{x}$) = 2

expand the expression:

• $log_{2}$(6x) = $log_{2}$ (6) + $log_{2}$ (x)

• $log_{2}$ ($\sqrt{x}$) = $log_{2}$ ($x^{\frac{1}{2} }$)

transform the expression:

• $log_{2}$ ($x^{\frac{1}{2} }$) = $\frac{1}{2}$ * $log_{2}$ (x)

calculate the difference:

• $log_{2}$ (x) - $\frac{1}{2}$ * $log_{2}$ (x) = $\frac{1}{2}$ * $log_{2}$ (x)

multiply both sides by 2:

• $log_{2}$ (6) + $\frac{1}{2}$ * $log_{2}$ (x) = 2   is now   2 $log_{2}$ (6) + $log_{2}$ (x) = 4

transform the expression:

• 2 $log_{2}$ (6) = $log_{2}$ ($6^{2}$)

simplify the expression:

• $log_{2}$ ($6^{2}$) + $log_{2}$ (x) = $log_{2}$ ($6^{2} x$)

evaluate the power:

• $log_{2}$ ($6^{2} x$) = $log_{2} (36x)$

convert the logarithm into exponential form:

• $log_{2}(36x) = 4$    is now   $36x = 2^{4}$

evaluate the power:

• $2^{4} = 16$

divide both sides by 36:

• $\frac{36}{36}$ x  =  $\frac{16}{36}$   which is   x = $\frac{4}{9}$