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Logarithms | Logarithms Exercise 9.2

Question 18
  1. Solve for x:

(i) log x + log 5 = 2 log 3

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(i) log x + log 5 = 2 log 3

Let us solve for x,

Log x = 2 log 3 – log 5

∴ x = 9/5

\begin{aligned} &\text { (ii) } \log _{3} x-\log _{3} 2=1\\ &\text { Let us solve for } x \text { , } \end{aligned}

\begin{aligned} \log _{3} x &=1+\log _{3} 2 \\ &=\log _{3} 3+\log _{3} 2\left[\text { since, } 1 \text { can be written as } \log _{3} 3=1\right] \\ &=\log _{3}(3 \times 2) \\ &=\log _{3} 6 \end{aligned}

\begin{aligned} &\text { (iii) } x=\log 125 / \log 25\\ &\begin{aligned} x &=\log 5^{3} / \log 5^{2} \\ &=3 \log 5 / 2 \log 5 \\ &=3 / 2[\text { since, } \log 5 / \log 5=1] \\ \therefore & x=3 / 2 \end{aligned} \end{aligned}

\begin{aligned} &\text { (iv) }(\log 8 / \log 2) \times(\log 3 / \log \sqrt{3})=2 \log x\\ &\begin{array}{l} \left(\log 2^{3} / \log 2\right) \times\left(\log 3 / \log 3^{1 / 2}\right)=2 \log x \\ (3 \log 2 / \log 2) \times(\log 3 / 1 / 2 \log 3)=2 \log x \\ 3 \times 1 /(1 / 2)=2 \log x \\ 3 \times 2=2 \log x \\ 6=2 \log x \\ \log x=6 / 2 \\ \log x=3 \\ x=(10)^{3} \\ =1000 \\ \therefore x=1000 \end{array} \end{aligned}

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Question 18

  1. Solve for x:

(i) log x + log 5 = 2 log 3

Looking to do well in your science exam ? Learn from an expert tutor. Book a free class!

(i) log x + log 5 = 2 log 3

Let us solve for x,

Log x = 2 log 3 – log 5

∴ x = 9/5

\begin{aligned} &\text { (ii) } \log _{3} x-\log _{3} 2=1\\ &\text { Let us solve for } x \text { , } \end{aligned}

\begin{aligned} \log _{3} x &=1+\log _{3} 2 \\ &=\log _{3} 3+\log _{3} 2\left[\text { since, } 1 \text { can be written as } \log _{3} 3=1\right] \\ &=\log _{3}(3 \times 2) \\ &=\log _{3} 6 \end{aligned}

\begin{aligned} &\text { (iii) } x=\log 125 / \log 25\\ &\begin{aligned} x &=\log 5^{3} / \log 5^{2} \\ &=3 \log 5 / 2 \log 5 \\ &=3 / 2[\text { since, } \log 5 / \log 5=1] \\ \therefore & x=3 / 2 \end{aligned} \end{aligned}

\begin{aligned} &\text { (iv) }(\log 8 / \log 2) \times(\log 3 / \log \sqrt{3})=2 \log x\\ &\begin{array}{l} \left(\log 2^{3} / \log 2\right) \times\left(\log 3 / \log 3^{1 / 2}\right)=2 \log x \\ (3 \log 2 / \log 2) \times(\log 3 / 1 / 2 \log 3)=2 \log x \\ 3 \times 1 /(1 / 2)=2 \log x \\ 3 \times 2=2 \log x \\ 6=2 \log x \\ \log x=6 / 2 \\ \log x=3 \\ x=(10)^{3} \\ =1000 \\ \therefore x=1000 \end{array} \end{aligned}

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