Final answer:
To solve the given logarithmic equation, we applied the property of logarithms that allows the difference of two logs to be written as the log of a quotient. We then converted the logarithmic equation to an exponential equation and solved for x, finding that x equals 4.
Step-by-step explanation:
To solve the equation log₂(12x−10)−log₂(4x+3)=1, we will first apply the properties of logarithms. Specifically, we can use the property that states the logarithm of a quotient is equal to the difference of the logarithms (logₙ(a) − logₙ(b) = logₙ(a/b)).
Therefore, we can rewrite the equation as:
log₂((12x−10)/(4x+3)) = 1
Now we can translate this logarithmic equation into its exponential form:
₂¹ = (12x−10)/(4x+3)
This simplifies to 2 = (12x−10)/(4x+3). Multiplying both sides by (4x+3) gives:
2(4x+3) = 12x − 10
Which simplifies to:
8x + 6 = 12x − 10
Subtracting 8x from both sides, we get:
6 = 4x − 10
Adding 10 to both sides, we find:
x = 4
Thus, the solution to the equation is x = 4.