Indices and Logarithms

3. Tutor-Marked Assignment

Section A: Indices (Laws of Exponents)

1. Simplify the following expressions:
   a) $3^4 \times 3^2$34×32
   b) $\frac{5^7}{5^4}$5457​
   c) $(2^3)^4$(23)4
   d) $8^{2/3}$82/3

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

3. Express in simplest form:
   a) $(a^3b^{-2})^4$(a3b−2)4
   b) $\frac{x^5 y^{-3}}{x^{-2} y^4}$x−2y4x5y−3​

Section B: Logarithms (Laws of Logarithms)

4. Evaluate:
   a) $\log_2 32$log2​32
   b) $\log_{10} 1000$log10​1000
   c) $\log_5 25 + \log_5 5$log5​25+log5​5

5. Solve for $x$x:
   a) $\log_3 x = 4$log3​x=4
   b) $\log_2 (x + 1) = 3$log2​(x+1)=3
   c) $\log_5 (x^2 - 4) = \log_5 5$log5​(x2−4)=log5​5

6. Express as a single logarithm:
   a) $\log_a x + \log_a y$loga​x+loga​y
   b) $2\log_b m - \frac{1}{2} \log_b n$2logb​m−21​logb​n

Bonus Question:

Prove that:
$\log_a (mn) = \log_a m + \log_a n$loga​(mn)=loga​m+loga​n
using the laws of logarithms.