Question 13 Marks
Solve for $x : \frac{\log 289}{\log 17}=\log x$
Answer$\frac{\log 289}{\log 17}=\log x $
$\Rightarrow \frac{\log 17^2}{\log 17}=\log x$
$ \Rightarrow \frac{2 \log 17}{\log 17}=\log x$
$ \Rightarrow 2=\log x $
$\Rightarrow 2 \log 10=\log x \ldots($ since $\log 10=1) $
$ \Rightarrow \log 10^2=\log x $
$\therefore x=10^2 $
$=100 .$
View full question & answer→Question 23 Marks
Solve for $x : \frac{\log 1331}{\log 11}=\log x$
Answer$\frac{\log 1331}{\log 11}=\log x$
$ \Rightarrow \frac{\log 11^3}{\log 11}=\log x$
$ \Rightarrow \frac{3 \log 11}{\log 11}=\log x $
$ \Rightarrow 3=\log x$
$ \Rightarrow 3 \log 10=\log x \ldots($ since $ \log 10=1)$
$ \Rightarrow \log 10^3=\log x $
$\therefore x =10^3 $
$=1000 . $
View full question & answer→Question 33 Marks
Solve for $x : \frac{\log 128}{\log 32}= x$
Answer$\frac{\log 128}{\log 32}=x$
$ \Rightarrow \frac{\log 2^7}{\log 2^5}=x$
$\Rightarrow \frac{7 \log 2}{5 \log 2}=x $
$ \Rightarrow x=\frac{7}{5} $
$ =1.4$
View full question & answer→Question 43 Marks
Solve for $x : \frac{\log 125}{\log 5}=\log x$
Answer$\frac{\log 125}{\log 5}=\log x$
$ \Rightarrow \frac{\log 5^3}{\log 5}=\log x $
$ \Rightarrow \frac{3 \log 5}{\log 5}=\log x $
$\Rightarrow 3=\log x $
$ \Rightarrow 3 \log 10=\log x \ldots($ since $\log 10=1) $
$ \Rightarrow \log 10^3=\log x$
$\therefore x=10^3 $
$= 1000 .$
View full question & answer→Question 53 Marks
Solve for $x : \frac{\log 121}{\log 11}=\log x$
Answer$\frac{\log 121}{\log 11}=\log x $
$ \Rightarrow \frac{\log 11^2}{\operatorname{lo11}}=\log x $
$ \Rightarrow \frac{2 \log 11}{\log 11}=\log x$
$=2=\log x$
$ \Rightarrow 2 \log 10=\log x \ldots($ since $\log 10=1) $
$ \Rightarrow \log 10^2=\log x$
$ \therefore x=10^2 $
$ =100 .$
View full question & answer→Question 63 Marks
Solve the following:$\log (x + 1) + \log (x - 1) = \log 48$
Answer$\log (x + 1) + \log (x - 1) = \log 48$
$\Rightarrow \log {(x + 1)(x - 1)} = \log 48$
$\Rightarrow \log (x^2 - 1) = \log 48$
$\Rightarrow x^2- 1 = 48$
$\Rightarrow x^2 = 49$
$\Rightarrow x = 7 ...($neglecting the negative value$).$
View full question & answer→Question 73 Marks
Solve the following:$\log _8 (x^2- 1) - \log _8 (3x + 9) = 0$
Answer$\log _8\left(x^2-1\right)-\log _8(3 x+9)=0$
$\Rightarrow \log _8\left(\frac{x^2-1}{3 x+9}\right)=\log _8 1$
$\Rightarrow \frac{x^2-1}{3 x+9}=1$
$\Rightarrow x^2-1=3 x+9$
$\Rightarrow x^2-3 x-10=0$
$\Rightarrow x^2-5 x+2 x-10=0$
$\Rightarrow x(x-5)+2(x-5)=0$
$\Rightarrow(x-5)(x+2)=0$
$\Rightarrow x=5 \text { or } x=-2$
View full question & answer→Question 83 Marks
Solve the following:$\log ( x + 1) + \log ( x - 1) = \log 11 + 2 \log 3$
Answer$\log ( x + 1) + \log ( x - 1) = \log 11 + 2 \log 3$
$\Rightarrow \log [(x + 1)(x - 1)] = \log 11 + \log 3^2$
$\Rightarrow \log {x^2 - 1} = \log (11.9)$
$\Rightarrow \log {x^2 - 1} = \log99$
$\Rightarrow x^2 - 1 = 99$
$\Rightarrow x^2 = 100$
So, $x = 10$ or $-10$
Negative value is rejected
So, $x = 10.$
View full question & answer→Question 93 Marks
Solve the following$:\log 7 + \log (3x - 2) = \log (x + 3) + 1$
Answer$\log 7+\log (3 x-2)=\log (x+3)+1$
$\Rightarrow \log 7+\log (3 x-2)-\log (x+3)=1$
$\Rightarrow \log \frac{7 \cdot(3 x-2)}{x+3}=\log 10$
$\Rightarrow \frac{7 \cdot(3 x-2)}{x+3}=10$
$\Rightarrow 21 x-14=10(x+3)$
$\Rightarrow 21 x-10 x=30+14$
$\Rightarrow 11 x=44$
$\Rightarrow x=\frac{44}{ 11}=4 .$
View full question & answer→Question 103 Marks
Solve the following:$\log(x^2+ 36) - 2\log x = 1$
Answer$\log \left(x^2+36\right)-2 \log x=1$
$\Rightarrow \log \left(x^2+36\right)-\log x 2=1$
$\Rightarrow \log \left(\frac{x^2+36}{x^2}\right)=1$
$=\log 10$
$\Rightarrow\left(\frac{x^2+36}{x^2}\right)=10$
$\Rightarrow x^2+36=10 x^2$
$\Rightarrow 9 x^2=36$
$\Rightarrow x^2=4$
$\Rightarrow x=2 .$
View full question & answer→Question 113 Marks
Solve the following:$\log (3 - x) - \log (x - 3) = 1$
Answer$\log (3-x)-\log (x-3)=1 $
$ \Rightarrow \log \left(\frac{3-x}{x-3}\right) $
$ =1 $
$=\log 10$
$ \Rightarrow\left(\frac{3-x}{x-3}\right)=10 $
$\Rightarrow 3-x=10(x-3) $
$ \Rightarrow 3-x=10 x-30$
$ \Rightarrow 11 x=33$
$ \Rightarrow x=3 .$
View full question & answer→Question 123 Marks
Express the following as a single logarithm:$\log \frac{81}{8}-2 \log \frac{3}{5}+3 \log \frac{2}{5}+\log \frac{25}{9}$
Answer$\log \frac{81}{8}-2 \log \frac{3}{5}+3 \log \frac{2}{5}+\log \frac{25}{9}$
$ =\log \frac{3^4}{2^3}-2 \log \frac{3}{5}+3 \log \frac{2}{5}+\log \frac{5^2}{3^2}$
$ =\log 3^4-\log 2^3-2 \log 3+2 \log 5+3 \log 2-3 \log 5+\log 5^2-\log 3^2$
$=4 \log 3-3 \log 2-2 \log 3+2 \log 5+3 \log 2-3 \log 5+2 \log 5- 2 \log 3$
$ =\log 5 .$
View full question & answer→Question 133 Marks
Express the following as a single logarithm:$2 \log \frac{15}{18}-\log \frac{25}{162}+\log \frac{4}{9}$
Answer$2 \log \frac{15}{18}-\log \frac{25}{162}+\log \frac{4}{9} $
$ 2 \log \frac{5}{2 \times 3}-\log \frac{5^2}{2 \times 3^4}+\log \frac{2^2}{3^2} $
$ =2 \log 5-2 \log 2-2 \log 3-\left\{\log 5^2-\log 2-\log 3^4\right\}+\log 2^2- \log 3^2 $
$=2 \log 5-2 \log 2-2 \log 3-2 \log 5+\log 2+4 \log 3+2 \log 2- 2 \log 3 $
$ =\log 2 .$
View full question & answer→Question 143 Marks
Express the following as a single logarithm:$\log 18 + \log 25 - \log 30$
Answer$\log 18 + \log 25 - \log 30$
$= \log (2 x 3^2) + \log 5^2 - \log (2 \times 3 \times 5)$
$= \log 2 + \log 3^2 + 2 \log 5 - {\log 2 + \log 3 + \log 5}$
$= \log 2 + 2 \log 3 + 2 \log 5 - \log 2 - \log 3 - \log 5$
$= \log 3 + \log 5$
$= \log (3 \times 5)$
$= \log 15.$
View full question & answer→Question 153 Marks
Write the logarithmic equation for:$V =\frac{4}{3} \pi r ^3$
Answer$V =\frac{4}{3} \pi r ^3$
Considering $\log$ on both the sides, we get
$\log V=\log \left(\frac{4}{3} \pi r^3\right)$
$=\log 4+\log \pi+\log r^3-\log 3$
$=\log 2^2+\log \pi+3 \log r-\log 3 $
$=2 \log 2-\log 3+\log \pi+3 \log r$
View full question & answer→Question 163 Marks
Write the logarithmic equation for:$E=\frac{1}{2} m v^2$
Answer$E =\frac{1}{2} m v ^2$
Considering $\log$ on both the sides, we get
$\log E=\log \left(\frac{1}{2} mv^2\right)$
$=\log \frac{1}{2}+\log m+\log v^2 $
$=\log 1-\log 2+\log m+2 \log v $
$=\log m+2 \log v-\log 2$
View full question & answer→Question 173 Marks
Write the logarithmic equation for:$F = G \frac{ m _1 m _2^2}{ d ^2}$
Answer$F=G \frac{m_1 m_2}{d^2}$
Considering $\log$ on both the sides, we get
$\log F=\log \left(G \frac{m_1 m_2}{d^2}\right)$
$=\log \left(G m_1 m_2\right)-\log d^2$
$=\log G+\log m_1+\log m_2-2 \log d$
View full question & answer→Question 183 Marks
If $\log (a + 1) = \log (4a - 3) - \log 3;$ find $a.$
Answer$\log (a + 1) = \log (4a - 3) - \log 3$
$\Rightarrow \log (a + 1) + \log 3 = \log (4a - 3)$
$\Rightarrow \log {3(a + 1)} = \log (4a - 3)$
$\Rightarrow 3 (a + 1) = 4a - 3$
$\Rightarrow 3a + 3 = 4a - 3$
$\Rightarrow 4a - 3a = 3 + 3$
$\Rightarrow a = 6.$
View full question & answer→Question 193 Marks
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. }$
View full question & answer→Question 203 Marks
Prove that $\log (1 + 2 + 3) = \log 1 + \log 2 + \log 3.$ Is it true for any three numbers $x, y, z$?
Answer$\log (1 + 2 + 3) = \log 6$
$= \log (1 + 2 + 3) = \log 1 + \log 2 + \log 3$
No, this property is not true for any numbers $x, y, z$
For example$, \log (1 + 3 + 5) = \log 9$
$\log 1 + \log 3 + \log 5 = \log (1 \times 3 \times 5) = \log 15$
$\log (1 + 3 + 5) \neq \log 1 + \log 3 + \log 5.$
View full question & answer→Question 213 Marks
Express the following in a form free from logarithm:$5 \log m - 1 = 3 \log n$
Answer$5 \log m-1=3 \log n$
$\Rightarrow \log m^5-\log 10=\log n^3 $
$ \Rightarrow \log \left(\frac{m^5}{10}\right)=\log n^3$
$ \Rightarrow\left(\frac{m^5}{10}\right)=n^3$
$ \Rightarrow m^5=10 n^3 .$
View full question & answer→Question 223 Marks
Express the following in a form free from logarithm:$m \log x - n \log y = 2 \log 5$
Answer$m \log x-n \log y=2 \log 5$
$ \Rightarrow \log x^m-\log y^n=\log 5^2$
$ \Rightarrow \log \left(\frac{x^m}{y^n}\right)=\log 5^2$
$\Rightarrow\left(\frac{x^m}{y^n}\right)=5^2=25 $
$ \Rightarrow x^m=25 y^n .$
View full question & answer→Question 233 Marks
Express the following in a form free from logarithm:$3 \log x - 2 \log y = 2$
Answer$3 \log x-2 \log y=2 $
$\Rightarrow \log x^3-\log y^2 $
$=2 \log 10$
$ \Rightarrow \log \left(\frac{x^3}{y^2}\right)=\log 10^2 $
$=\log 100$
$\Rightarrow\left(\frac{x^3}{y^2}\right)=100 $
$ \Rightarrow x^3=100 y^2$
View full question & answer→Question 243 Marks
Express the following in terms of $\log 2$ and $\log 3: \log \frac{26}{51}-\log \frac{91}{119}$
Answer$\log \frac{26}{51}-\log \frac{91}{119}$
$ =\log \frac{2 \times 13}{3 \times 17}-\log \frac{7 \times 13}{7 \times 17} $
$ =\log \frac{2 \times 13}{3 \times 17}-\log \frac{13}{17} $
$ =\log 13+\log 2-\log 3-\log 17-\log 13+\log 17 $
$=\log 2-\log 3 .$
View full question & answer→Question 253 Marks
Express the following in terms of $\log 2$ and $\log 3: \log \sqrt[5]{216}$
Answer$\log \sqrt[5]{216} $
$=\log (216)^{\frac{1}{5}}$
$ =\frac{1}{5} \log 216$
$ =\frac{1}{5} \log \left(2^3 \times 3^3\right) $
$ =\frac{1}{5} \log 2^3+\frac{1}{5} \log 3^3$
$ =\frac{3}{5} \log 2+\frac{3}{5} \log 3$
View full question & answer→Question 263 Marks
Express the following in terms of $\log 2$ and $\log 3: \log \sqrt[3]{144}$
Answer$\log \sqrt[3]{144}$
$=\log (144)^{\frac{1}{3}}$
$ =\frac{1}{3} \log 144$
$=\frac{1}{3} \log \left(2^4 \times 3^2\right) $
$=\frac{1}{3} \log 2^4+\frac{1}{3} \log 3^2 $
$ =\frac{4}{3} \log 2+\frac{2}{3} \log 3 .$
View full question & answer→Question 273 Marks
If $2 \log x + 1 =\log 360,$ find $: \log(2 x -2)$
Answer$\log (2 x-2) $
$2 \log x+1=\log 360 $
$\Rightarrow \log x^2+\log 10=\log 360 $
$\Rightarrow \log \left(10 x^2\right)=\log 360 $
$\Rightarrow 10 x^2=360 $
$\Rightarrow x^2=\frac{360}{10}=36 $
$\Rightarrow x=\sqrt{36}= \pm 6$
As negative value is rejected,
$\therefore x=6 $
$\therefore \log (2 x-2) $
$=\log (2 \cdot 6-2) $
$=\log 10 $
$=1 .$
View full question & answer→Question 283 Marks
If $2 \log x + 1 =\log 360,$ find$: x$
Answer$x$
$2 \log x+1=\log 360$
$\Rightarrow \log x^2+\log 10=\log 360$
$\Rightarrow \log \left(10 x^2\right)=\log 360$
$\Rightarrow 10 x^2=360$
$\Rightarrow x^2=\frac{360}{10}=36$
$\Rightarrow x=\sqrt{36}= \pm 6$
As negative value is rejected,
$\therefore x =6 \text {. }$
View full question & answer→Question 293 Marks
If $\log 27 = 1.431,$ find the value of the following$: \log 9$
Answer$\log 27$
$=\log 3^3$
$=3 \log 3$
$=1.431$
$\Rightarrow \log 3$
$=\frac{1.431}{3}$
$=0.477$
$\therefore \log 9$
$=\log 3^2$
$=2 \log 3$
$=2 \times 0.477$
$=0.954 .$
View full question & answer→Question 303 Marks
If $\log 8=0.90$, find the value of each of the following: $\log \sqrt{32}$
Answer$\log \sqrt{32} $
$ =\frac{1}{2} \log 32$
$ =\frac{1}{2} \log 2^5 $
$ =\frac{5}{2} \log 2$
$ =\frac{5}{2} \times 0.30 $
$=0.75 .$
View full question & answer→Question 313 Marks
If $\log 2 = x$ and $\log 3 = y,$ find the value of each of the following on terms of $x$ and $y: \log1.2$
Answer$\log 1.2$
$=\log \left(\frac{12}{10}\right)$
$=\log 12-\log 10$
$=\log \left(2^2 \times 3\right)-1$
$=\log 2^2+\log 3-1$
$=2 \log 2+\log 3-1$
$=2 x+y-1 .$
View full question & answer→Question 323 Marks
If $2 \log y - \log x - 3 = 0,$ express $x$ in terms of $y.$
Answer$2 \log -\log x-3=0$
$\Rightarrow \log x=2 \log y-3$
$\Rightarrow \log x=\log y^2-3 \log 10 \ldots[\because \log 10=1]$
$\Rightarrow \log x=\log y^2-\log 10^3$
$\Rightarrow \log x=\log \left(\frac{y^2}{1000}\right)$
$\therefore x=\frac{y^2}{1000} .$
View full question & answer→Question 333 Marks
If $\log 2=0.3010, \log 3=0.4771$ and $\log 5=0.6990$, find the values of: $\log 540$
Answer$\log540$
$= \log (2^2 \times 3^3 \times 5)$
$= \log 2^2 + \log 3^3+ \log 5$
$= 2 \log 2 + 3 \log 3 + \log 5$
$= (2 \times 0.3010) + (3 \times 0.4771) + 0.6990$
$= 2.7323.$
View full question & answer→Question 343 Marks
If $\log 2 = 0.3010, \log 3 = 0.4771$ and $\log 5 = 0.6990$, find the values of:$ \log45$
Answer$\log45$
$= \log (3^2 \times 5)$
$= \log 3^2 + \log 5$
$= 2 \log 3 + \log 5$
$= (2 \times 0.4771) + 0.6990$
$= 1.6532.$
View full question & answer→Question 353 Marks
If $\log 2=0.3010, \log 3=0.4771$ and $\log 5=0.6990$, find the values of: $\log 18$
Answer$\log18$
$= \log (2 \times 3^2)$
$= \log 2 + \log 3^2$
$= \log 2 + 2 \log 3$
$= 0.3010 + (2 \times 0.4771)$
$= 1.2552.$
View full question & answer→Question 363 Marks
If $\log _3 m= x$ and $\log _3 n = y$, write down $3^{2 x-3}$ in terms of $m$
Answer$3^{2 x-3}$ in terms of $m$
$\log _3 m - x$
$\Rightarrow m =3^{ x }$
$\therefore 3^{2 x -3}$
$=\frac{3^2 x}{3^3}$
$=\frac{\left(3^x\right)^2}{27}$
$=\frac{ m ^2}{27}$
View full question & answer→Question 373 Marks
Express the following in terms of $\log 2$ and $\log 3: \log 12^8$
Answer$\log12^8$
$=\log (3 \times 2^2)^8$
$= 8\log (3 \times 2^2)$
$= 8[\log 3 + \log2^2]$
$= 8[\log 3 + 2\log 2].$
View full question & answer→Question 383 Marks
If $\log x=a$ and $\log y=b$, write down $10^{2 b}$ in terms of $y$
Answer$10^{2b}$ in terms of $y$
$\log y = b$
$\Rightarrow y = 10^b$
$\therefore 10^{2b}$
$= (10^b)^2$
$= y^2.$
View full question & answer→Question 393 Marks
If $\log x=A+B$ and $\log y=A-B$, express the value of $\log \frac{x^2}{10 y}$ in terms of $A$ and $B$.
Answer$\log \frac{x^2}{10 y} $
$ =\log x^2-\log 10 y$
$=2 \log x-(\log 10+\log y) $
$=2 \log x-\log y-1 $
$ =2(A+B)-(A-B)-1 $
$ =A+3 B-1 .$
View full question & answer→Question 403 Marks
If $\log x=p+q$ and $\log y=p-q$, find the value of $\log \frac{10 x}{y^2}$ in terms of $p$ and $q$.
Answer$\log x=p+q$ and $\log y=p-q $
$ \log \frac{10 x}{y^2}=\log 10 x-\log y^2 $
$\Rightarrow \log \frac{10 x}{y^2}=\log 10+\log x-2 \log y $
$ \Rightarrow \log \frac{10 x}{y^2}=1+p+q-2(p-q) $
$\Rightarrow \log \frac{10 x}{y^2}=1-p+3 q .$
View full question & answer→Question 413 Marks
Express the following in terms of $\log 2$ and $\log 3: \log 54$
Answer$\log 54$
$= \log (2 \times 3 \times 3 \times 3)$
$= \log (2 \times 3^3)$
$= \log 2 + \log 3^3$
$= \log 2 + 3 \log 3.$
View full question & answer→Question 423 Marks
Express the following in terms of $\log 2$ and $\log 3: \log 36$
Answer$\log 36$
$= \log (2 \times 2 \times 3 \times 3)$
$= \log (2^2 \times 3^2)$
$= \log 2^2 + \log 3^2$
$= 2\log 2 + 2\log 3.$
View full question & answer→Question 433 Marks
If $\log _{10} a = x$, and $\log _{10} c = z$, find $10^{x-y+z}$ in terms of $a , b$ and $c$
Answer$10^{x-y+z}$ in terms of $a , b$ and $c . $
$ \log _{10} a = x$
$ \Rightarrow a =10^{ x }$
$ \log _{10} b = y $
$ \Rightarrow b =10^{ y }$
$ \log _{10} c = z $
$ \Rightarrow c =10^{ z }$
$ 10^{x-y+z} $
$ =10^{ x } \cdot 10^{-y} \cdot 10^{ z }$
$ = a \cdot b ^{-1} \cdot c $
$ =\frac{ ac }{ b } \cdot$
View full question & answer→Question 443 Marks
If $\log _{10} x=a, \log _{10} y=b$ and $\log _{10} z=2 a-3 b$, express $z$ in terms of $x$ and $y$.
Answer$\log _{10} x=a$
$\Rightarrow x=10^a$
$\log _{10} y=b$
$\Rightarrow y=10^b$
$\log _{10} z=2 a-3 b$
$\Rightarrow z=10^{2 a-3 b}$
$\therefore z=10^{2 a-3 b}$
$=\left(10^a\right)^2 \cdot\left(10^b\right)^{-3}$
$=(x)^2(y)^{-3}$
$=\frac{x^2}{y^3} .$
View full question & answer→Question 453 Marks
If $\log _{10} x = a$, express the following in terms of $x: 10 ^{a + 3}$
Answer$10 ^{a + 3}$
$\log _{10} x = a$
$\Rightarrow x = 10^a$
$\therefore 10 ^{a + 3}$
$= 10 ^a . 10^3$
$= x .1000$
$= 1000x.$
View full question & answer→Question 463 Marks
Find the value of: $\log _{\sqrt{3}}(3 \sqrt{3})$
Answer$\log _{\sqrt{3}}(3 \sqrt{3}) $
Let $\log _{\sqrt{3}}(3 \sqrt{3})= x$
$ \Rightarrow(\sqrt{3})^x=3 \sqrt{3}$
$ \Rightarrow 3^{\frac{x}{2}}=3^{1+\frac{1}{2}}=3^{\frac{3}{2}} $
$ \therefore \frac{x}{2}=\frac{3}{2} $
$ \Rightarrow x =3 .$
View full question & answer→Question 473 Marks
Find the value of: $\log _{0.01}10$
Answer$\log _{0.01} 10$
Let $\log _{0.01} 10=x$
$\Rightarrow(0.01)^x=10$
$\Rightarrow(10-2)^x=10^1$
$\Rightarrow 10^{-2 x}=10^1$
$\Rightarrow x=-\frac{1}{2}$
$\therefore-2 x=1 .$
View full question & answer→Question 483 Marks
Find the value of: $\log _8 2$
Answer$\log _8 2$
Let ${\log _8 2=x}$
$\Rightarrow 2=8^x$
$\Rightarrow\left(2^3\right)^x=2$
$\Rightarrow 2^{3 x}=2^1$
$\Rightarrow x=\frac{1}{3}$
$\therefore 3 x=1 .$
View full question & answer→