Solve the Logarithmic Equation: log4x + log2 - log9 = log₂4

$\log4x+\log2-\log9=\log_24$

?=x

To solve the equation $\log 4x + \log 2 - \log 9 = \log_2 4$, we will follow these steps:

• Step 1: Simplify the left side using logarithmic properties
• Step 2: Convert the right side using change of base
• Step 3: Equate the simplified expressions and solve for $x$

Step 1: Simplify the left side:

The left side $\log 4x + \log 2 - \log 9$ can be combined using the properties of logarithms:

$\log 4x + \log 2 = \log(4x \cdot 2) = \log(8x)$

Now, using the subtraction property:

$\log (8x) - \log 9 = \log \left(\frac{8x}{9}\right)$

Step 2: Convert the right side using the change of base formula:

$\log_2 4 = \frac{\log 4}{\log 2}$

We recognize that $4 = 2^2$, so $\log_2 4 = 2$.

Step 3: Equate the expressions and solve for $x$:

Now equate:

$\log \left(\frac{8x}{9}\right) = 2$

This implies:

$\frac{8x}{9} = 10^2 = 100$

Thus, solving for $x$:

$8x = 900$

$x = \frac{900}{8} = 112.5$

Therefore, the solution to the problem is $x = 112.5$.