Logbase3(4X +1) -Logbase3(3X -5) =2
To solve the equation Logbase3(4X + 1) - Logbase3(3X - 5) = 2, we can use the properties of logarithms.
Step 1: Combine the logarithms using the quotient rule. According to the quotient rule of logarithms, LogbaseA(B) - LogbaseA(C) = LogbaseA(B / C). Applying this rule, we have:
Logbase3((4X + 1) / (3X - 5)) = 2
Step 2: Rewrite the equation in exponential form. In exponential form, the equation will be:
3^2 = (4X + 1) / (3X - 5)
Step 3: Simplify the equation. Square 3:
9 = (4X + 1) / (3X - 5)
Step 4: Cross-multiply to eliminate the fractions:
9 * (3X - 5) = 4X + 1
27X - 45 = 4X + 1
Step 5: Combine like terms:
27X - 4X = 1 + 45
23X = 46
Step 6: Solve for X:
X = 46 / 23
X = 2
So, the solution to the equation Logbase3(4X + 1) - Logbase3(3X - 5) = 2 is X = 2.
To solve the given equation, we'll use logarithmic properties and algebraic manipulation. Here's how we can proceed:
Step 1: Apply logarithmic properties
Using the property of logarithms, we can rewrite the equation as follows:
logbase3(4X + 1) - logbase3(3X - 5) = 2
Step 2: Combine the logarithmic terms
Using the property of logarithmic subtraction, the equation can be further simplified:
logbase3[(4X + 1) / (3X - 5)] = 2
Step 3: Rewrite the equation in exponential form
Converting the equation back to exponential form will help us solve for X. In exponential form, the base 3 is raised to the power of 2, equal to the expression inside the logarithm:
3^2 = (4X + 1) / (3X - 5)
Step 4: Solve the equation
Multiply both sides of the equation by (3X - 5) to eliminate the fraction:
9(3X - 5) = 4X + 1
Expanding and simplifying the equation:
27X - 45 = 4X + 1
Combine like terms:
27X - 4X = 1 + 45
23X = 46
Divide both sides by 23 to isolate X:
X = 46 / 23
Simplifying the fraction:
X = 2
Therefore, the value of X that satisfies the equation logbase3(4X + 1) - logbase3(3X - 5) = 2 is X = 2.
Let's solve this logarithmic equation step by step.
Using the logarithmic properties:
1. Logbase3(4X + 1) - Logbase3(3X - 5) = Logbase3((4X + 1)/(3X - 5))
Rearranging the equation:
2 = Logbase3((4X + 1)/(3X - 5))
Now, we can rewrite the equation in exponential form:
3^2 = (4X + 1)/(3X - 5)
9 = (4X + 1)/(3X - 5)
Cross-multiplying:
9(3X - 5) = 4X + 1
27X - 45 = 4X + 1
Subtracting 4X from both sides:
27X - 4X - 45 = 1
Simplifying the equation:
23X - 45 = 1
Adding 45 to both sides:
23X = 46
Dividing both sides by 23:
X = 2
Therefore, the solution to the equation is X = 2.