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Exponential equation a^f(x) = b (products of powers): take logs to solve for x

This type of question, asked 4 times: CDS 2 2016, CDS 2 2020, CDS 1 2021, CDS 2 2021. All 4 questions below, oldest first.

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  1. Question 1

    CDS 2 2016

    28
    What are the roots of the equation 2x+2⋅27xx−1=92^{x+2} \cdot 27^{\frac{x}{x-1}} = 9 ?

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  2. Question 2

    CDS 2 2020

    43
    If 63−4x 4x+5=86^{3 - 4x}\, 4^{x + 5} = 8 (Given log⁡102=0.301\log_{10} 2 = 0.301 and log⁡103=0.477\log_{10} 3 = 0.477), then which one of the following is correct ?

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  3. Question 3

    CDS 1 2021

    45
    If 5x−3=85^{x-3} = 8, then what is xx equal to?

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  4. Question 4

    CDS 2 2021

    20
    If 5x−1=(2.5)log⁡1055^{x-1}=(2.5)^{\log_{10}5}, then what is the value of xx ?

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