3. Tutor-Marked Assignment

Section A: Indices (Laws of Exponents)

  1. Simplify the following expressions:
    a) 34×323^4 \times 3^2
    b) 5754\frac{5^7}{5^4}
    c) (23)4(2^3)^4
    d) 82/38^{2/3}

  2. Solve for xx in the following equations:
    a) 2x=322^x = 32
    b) 5x+1=1255^{x+1} = 125
    c) 9x−1=27x+29^{x-1} = 27^{x+2}

  3. Express in simplest form:
    a) (a3b−2)4(a^3b^{-2})^4
    b) x5y−3x−2y4\frac{x^5 y^{-3}}{x^{-2} y^4}

Section B: Logarithms (Laws of Logarithms)

  1. Evaluate:
    a) log⁡232\log_2 32
    b) log⁡101000\log_{10} 1000
    c) log⁡525+log⁡55\log_5 25 + \log_5 5

  2. Solve for xx:
    a) log⁡3x=4\log_3 x = 4
    b) log⁡2(x+1)=3\log_2 (x + 1) = 3
    c) log⁡5(x2−4)=log⁡55\log_5 (x^2 - 4) = \log_5 5

  3. Express as a single logarithm:
    a) log⁡ax+log⁡ay\log_a x + \log_a y
    b) 2log⁡bm−12log⁡bn2\log_b m - \frac{1}{2} \log_b n

Bonus Question:

Prove that:
log⁡a(mn)=log⁡am+log⁡an\log_a (mn) = \log_a m + \log_a n
using the laws of logarithms.