Question
Prove that $(\log a)^2-(\log b)^2=\log \left(\frac{a}{b}\right) \cdot \log (a b)$
✓
Answer
$\text { L.H.S. } $
$ =(\log a)^2-(\log b)^2$
$=(\log a+\log b)(\log a-\log b) \quad \ldots\left\{\text { using identity } m^2-n^2=(m+n)(m-n)\right\} $
$=\log (a b) \log \left(\frac{a}{b}\right) $
$ =\log \left(\frac{a}{b}\right) \cdot \log (a b)$
$ =\text { R.H.S. }$
$ =(\log a)^2-(\log b)^2$
$=(\log a+\log b)(\log a-\log b) \quad \ldots\left\{\text { using identity } m^2-n^2=(m+n)(m-n)\right\} $
$=\log (a b) \log \left(\frac{a}{b}\right) $
$ =\log \left(\frac{a}{b}\right) \cdot \log (a b)$
$ =\text { R.H.S. }$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Factorise : $100-9 p^2$
→Factorise the following:$2 \sqrt{5} x^2-7 x-3 \sqrt{5}$
→Find the area of a rhombus one side of which measures 20 cm and one of whose diagonals is 24 cm.
A sheet is 11 cm long and 2 cm wide. Circular pieces 0.5 cm in diameter are cut from it to prepare discs. Calculate the number of discs that can be prepared.
[Hint: Each circular dise will be cut from a square sheet of side 0.5 сm.
$\therefore$ Number of circular discs = Number of square sheets each of side 0.5 сm
$\begin{array}{l}=\frac{\text { Area of given sheet }}{\text { Area of each square sheet }} \\ \left.=\frac{(11 \times 2)}{\left(\frac{1}{2} \times \frac{1}{2}\right)}=(11 \times 8)=88 .\right]\end{array}$
→[Hint: Each circular dise will be cut from a square sheet of side 0.5 сm.
$\therefore$ Number of circular discs = Number of square sheets each of side 0.5 сm
$\begin{array}{l}=\frac{\text { Area of given sheet }}{\text { Area of each square sheet }} \\ \left.=\frac{(11 \times 2)}{\left(\frac{1}{2} \times \frac{1}{2}\right)}=(11 \times 8)=88 .\right]\end{array}$
Find three rational numbers between: $\frac{1}{2}$ and $\frac{3}{5}$
→Factorise the following by splitting the middle term:$12 + x - 6x^2$
→If $\log _{10} x=a, \log _{10} y=b$ and $\log _{10} z=2 a-3 b$, express $z$ in terms of $x$ and $y$.
→In the given figure, $AD = AE$ and $\angle BAD =\angle CAE$. Prove that: $AB = AC$.
→Represent each of the following on the number line :
(i) $\frac{3}{7}$$\quad$(ii) $\frac{16}{5}$$\quad$(iii) $-\frac{4}{9}$$\quad$(iv) $-\frac{18}{11}$$\quad$(v) $-3 \frac{1}{6}$
→(i) $\frac{3}{7}$$\quad$(ii) $\frac{16}{5}$$\quad$(iii) $-\frac{4}{9}$$\quad$(iv) $-\frac{18}{11}$$\quad$(v) $-3 \frac{1}{6}$
A toothed wheel of diameter 50 cm is attached to a smaller wheel of diameter 30 cm. How many revolutions will the smaller wheel make when the larger one makes 30 revolutions?
[Hint. Required number of revolutions
$\left.=\frac{\text { Distance moved by toothed wheel in } 30 \text { revolutions }}{\text { Distance moved by smaller wheel in } 1 \text { revolution }} \cdot\right]$
→[Hint. Required number of revolutions
$\left.=\frac{\text { Distance moved by toothed wheel in } 30 \text { revolutions }}{\text { Distance moved by smaller wheel in } 1 \text { revolution }} \cdot\right]$