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(iii) Log_(4)(3^(x)-1)*log_(1/4)((3^(x)-1)/(16))

3 min read Jun 16, 2024
(iii) Log_(4)(3^(x)-1)*log_(1/4)((3^(x)-1)/(16))

Simplifying the Expression: log_4(3^x - 1) * log_(1/4)((3^x - 1)/16)

This expression involves logarithms with different bases, making it a bit tricky to work with directly. To simplify it, we'll use some key properties of logarithms:

Key Properties of Logarithms

  1. Change of Base Formula: log_a(b) = log_c(b) / log_c(a) (where c is any positive number different from 1)

  2. Logarithm of a Quotient: log_a(b/c) = log_a(b) - log_a(c)

  3. Logarithm of a Power: log_a(b^c) = c * log_a(b)

Simplifying the Expression

Let's apply these properties to simplify the given expression step by step:

  1. Change of Base for the Second Logarithm:

    We'll change the base of the second logarithm to 4, using the change of base formula:

    log_(1/4)((3^x - 1)/16) = log_4((3^x - 1)/16) / log_4(1/4)

    Since log_4(1/4) = -1, we have:

    log_(1/4)((3^x - 1)/16) = -log_4((3^x - 1)/16)

  2. Simplifying the Second Logarithm:

    Using the logarithm of a quotient property:

    -log_4((3^x - 1)/16) = -[log_4(3^x - 1) - log_4(16)]

    We know log_4(16) = 2 (since 4^2 = 16). Therefore:

    -log_4((3^x - 1)/16) = -log_4(3^x - 1) + 2

  3. Combining the Simplified Expressions:

    Now, we can substitute the simplified forms of both logarithms back into the original expression:

    log_4(3^x - 1) * log_(1/4)((3^x - 1)/16) = log_4(3^x - 1) * (-log_4(3^x - 1) + 2)

  4. Expanding the Expression:

    Expanding the product:

    log_4(3^x - 1) * (-log_4(3^x - 1) + 2) = -[log_4(3^x - 1)]^2 + 2 * log_4(3^x - 1)

Final Result:

The simplified form of the given expression is:

-[log_4(3^x - 1)]^2 + 2 * log_4(3^x - 1)

This is a quadratic expression in terms of log_4(3^x - 1).

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