Find y without using tables.
If 2 + log₂ 3 + log₂ y = log₂ 5 + 1
Answer (1)
To solve the equation 2 + lo g 2 3 + lo g 2 y = lo g 2 5 + 1 , we'll go through step-by-step.
First, let's rearrange the equation to group the logarithmic terms together: lo g 2 3 + lo g 2 y = lo g 2 5 + 1 − 2.
Simplifying the right side, we get: lo g 2 3 + lo g 2 y = lo g 2 5 − 1.
Recognize that 1 = lo g 2 2 , so it becomes: lo g 2 3 + lo g 2 y = lo g 2 5 − lo g 2 2.
Using the properties of logarithms, lo g b x − lo g b y = lo g b ( y x ) , so: lo g 2 3 + lo g 2 y = lo g 2 ( 2 5 ) .
On the left, use the property lo g b x + lo g b y = lo g b ( x y ) , giving: lo g 2 ( 3 y ) = lo g 2 ( 2 5 ) .
Since the logarithmic expressions are equal, their arguments must be equal: 3 y = 2 5 .
Solve for y by dividing both sides by 3: y = 2 5 × 3 1 = 6 5 .
Therefore, the value of y is 6 5 .
