Examples
Laws of Logs
- \(\log_{a}n=x\)
- \(a^x=n\)
- \(\log_{a}x+\log_{a}y=\log_{a}xy\)
- \(\log_{a}x-\log_{a}y=\log_{a} \frac{x}{y}\)
- \(\log_{a}(x^k)=k\log _a(x)\)
- \(\log_{a} \frac{1}{x}=\log_{a}x^{-1}=-\log _ax\)
- \(\log_{a}a=1\)
- \(\log_{a}1=0\)
Exercise 14B
Exercise 14B Q1
- 2.71828
- 54.59815
- 0.00004
- 1.22140
Exercise 14B Q2
Exercise 14B Q3
Exercise 14D
Exercise 14D Q1
- \(4^4=256\) is the same as \(\log_{4}256=4\)
- \(3^{-2}=\frac{1}{9}\) is the same as \(\log_{3} \frac{1}{9}=-2\)
- \(10^6=1000000\) is the same as \(\log 1000000=6\)
- \(11^1=11\) is the same as \(\log_{11}11=1\)
- \((0.2)^3=0.008\) is the same as \(\log_{0.2}0.008=3\)
Exercise 14D Q2
- \(\log_{2}16=4\) is the same as \(4^2=16\)
- \(\log_{5}25=2\) is the same as \(5^2=25\)
- \(\log_{9}3=\frac{1}{2}\) is the same as \(9^{\frac{1}{2}}=3\)
- \(\log_{5}0.2=-1\) is the same as \(5^{-1}=0.2\)
- \(\log 100000=5\) is the same as \(10^5=100000\)
Exercise 14D Q3
- \(\log_{2}8=3\)
- \(\log_{5}25=2\)
- \(\log 10000000=7\)
- \(\log_{12}12=1\)
- \(\log_{3}729=5\)
- \(\log_{10}\sqrt{ 10 }=\frac{1}{2}\)
- \(\log_{4}0.25=-1\)
- \(\log_{0.25}16=-2\)
- \(\log_{a}a^{10}=10\)
- \(\log_{\frac{2}{3}} \frac{9}{4}=2\)
Exercise 14D Q4
- \(\log_{5}x=4\) therefore \(x=5^4=625\)
- \(\log_{x}81=2\) therefore \(x=81^{\frac{1}{3}}=3\)
- \(\log_{7}x=1\) therefore \(x=7\)
- \(\log_{2}(x-1)=3\) therefore \(x=2^3+1=9\)
- \(\log_{3}(4x+1)=4\) therefore \(x=\frac{(3^4)-1}{4}=20\)
- \(\log_{x}2x=2\) therefore \(x=2\) because \(\log_{2}4=2\)
Exercise 14D Q5
- \(\log_{9}230=2.475\)
- \(\log_{5}33=2.173\)
- \(\log_{10}1020=3.009\)
Exercise 14E
Exercise 14E Q1
- \(\log_{2}7+\log_{2}3=\log_{2}21\)
- \(\log_{2}36-\log_{2}4=\log_{2}9\)
- \(3\log_{5}2+\log_{5}10=\log_{5}80\)
- \(2\log_{6}8-4\log_{6}3=\log_{6} \frac{64}{81}\)