(1)/(6)log_2)(x-2)-(1)/(3)=log_(1/8)sqrt(3x-5)

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Jun 07, 2024

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Solving the Equation: (1/6)log‚2)(x-2) - (1/3) = log‚(1/8)√(3x-5)

In this article, we will explore the solution to the equation (1/6)log‚2)(x-2) - (1/3) = log‚(1/8)√(3x-5). This equation involves logarithmic functions and requires some manipulation to solve.

Step 1: Simplify the Equation

First, let's simplify the equation by combining like terms:

(1/6)log‚2)(x-2) - (1/3) = log‚(1/8)√(3x-5)

(1/6)log‚2)(x-2) - 1/3 = log‚(1/8)√(3x-5)

Step 2: Use Logarithmic Properties

Next, we can use the logarithmic property that states log‚a)M = N is equivalent to a^N = M. Applying this property to our equation, we get:

2 ^ ((1/6)(x-2) - (1/3)) = (√(3x-5))^3

Step 3: Simplify Further

Simplifying the equation further, we get:

2 ^ ((x-2)/6 - 2/3) = (√(3x-5))^3

2 ^ ((x-2)/6 - 2/3) = (3x-5)^(3/2)

Step 4: Solve for x

Now, we can solve for x by applying the exponential function to both sides of the equation:

((x-2)/6 - 2/3) = log‚2)(3x-5)^(3/2)

x - 2 = 6(log‚2)(3x-5)^(3/2) + 4

x = 6(log‚2)(3x-5)^(3/2) + 6

Conclusion

In this article, we have successfully solved the equation (1/6)log‚2)(x-2) - (1/3) = log‚(1/8)√(3x-5) using logarithmic properties and exponential functions. The final solution is x = 6(log‚2)(3x-5)^(3/2) + 6.