Exercises 7 (Logarithms)

  1. Determine the values for the following logarithms:

    1. \(\log_2 98.5\)
    2. \(\log_4 22.86\)
    3. \(\log_7 1050\)
    4. \(\log_8 211.746\)

  2. The formula \(\dfrac{T_1}{T_2}=e^{\mu\theta}\) shows the relationship between the tension in a flat driving belt (\(T_1\) being on the tight or driving side) whilst \(\theta\) is the angle of contact around the pulley and \(\mu\) is the coefficient of friction.

    1. Show that \(\log T_1 = \log T_2 + \mu\theta\log e\)
    2. Determine \(T_1\) using logs if \(T_2=150\), \(\mu=0.25\), \(\theta=3\) and \(e=2.718\).

  3. Solve for \(x\):

    1. \(5^{2x}=0.5\)
    2. \(3^{2x}+3^x=12\)
    3. \(3^{2x}-3^{x+1}+2=0\)

  4. If \(10^x+10^{-x}=4\), show that \(x=\log_{10}(2\pm\sqrt{3})\).

  5. Express in terms of \(\log p\), \(\log q\) and \(\log r\):

    1. \(\log pq\)
    2. \(\log pqr\)
    3. \(\log pq/r\)
    4. \(\log p/qr\)
    5. \(\log p^2q\)
    6. \(\log q/r^2\)
    7. \(\log p^2q^3/r\)
    8. \(\log p^nq^m\)
    9. \(\log 2pq\)
    10. \(\log 2pq^2\)

  6. Simplify:

    1. \(\log p+\log q\)
    2. \(2\log p+\log q\)
    3. \(\log q-\log r\)
    4. \(3\log q+4\log p\)
    5. \(\log p+2\log q-3\log r\)
    6. \(\log p-\log 2\)
    7. \(2\log p-p\log 2\)
    8. \(\log(p+2)-\log(q-2)\)

Solutions: 1. (i) \(6.62\); (ii) \(2.26\); (iii) \(3.57\); (iv) \(2.58\); 2. (ii) \(T_1=317\); 3. (i) \(x=-0.22\); (ii) \(x=1\); (iii) \(x=0.63\) or \(x=0\); 5. (a) \(\log p+\log q\); (b) \(\log p+\log q+\log r\); (c) \(\log p+\log q-\log r\); (d) \(\log p-\log q-\log r\); (e) \(2\log p+\log q\); (f) \(\log q-2\log r\); (g) \(2\log p+3\log q-\log r\); (h) \(n\log p+m\log q\); (i) \(\log 2+\log p+\log q\); (j) \(\log 2+\log p+2\log q\); 6. (a) \(\log pq\); (b) \(\log p^2q\); (c) \(\log q/r\); (d) \(\log q^3p^4\); (e) \(\log pq^2/r^3\); (f) \(\log p/2\); (g) \(\log p^2/2^p\); (h) \(\log (p+2)/(q-2)\)