Question

3/log4x-2 -2/log4x-1=2

Answer 1

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Answer 2

To solve the equation (3/log₄(x-2)) - (2/log₄(x-1)) = 2, we can follow these steps:

Step 1: Simplify the equation by finding a common denominator for the logarithms.

Multiply the first term by log₄(x-1) and the second term by log₄(x-2):

[3(log₄(x-1))/log₄(x-2)] - [2(log₄(x-2))/log₄(x-1)] = 2

Step 2: Simplify further.

Applying the properties of logarithms, we can rewrite the equation as:

[3log₄(x-1) - 2log₄(x-2)] / log₄(x-2) = 2

Step 3: Combine the logarithms.

Using the logarithmic property logₐ(b) - logₐ(c) = logₐ(b/c), we can rewrite the equation as:

log₄[(x-1)³/(x-2)²] / log₄(x-2) = 2

Step 4: Convert the equation to exponential form.

Rewriting the equation in exponential form gives us:

4² = [(x-1)³/(x-2)²]

Step 5: Solve for x.

16 = (x-1)³ / (x-2)²

Now, we can solve the equation. To do that, we can cross-multiply:

16(x-2)² = (x-1)³

Expand both sides:

16(x² - 4x + 4) = (x-1)(x-1)(x-1)

16x² - 64x + 64 = (x-1)³

16x² - 64x + 64 = (x³ - 3x² + 3x - 1)

Rearrange the terms to get a cubic equation:

x³ - 3x² + 3x - 1 - 16x² + 64x - 64 = 0

x³ - 19x² + 67x - 65 = 0

At this point, solving for x numerically or algebraically becomes quite complex. It may require approximation methods or using numerical software to find the approximate values of x that satisfy the equation.