Question 12 Marks
Express $\log_{10}3 + 1$ in terms of $\log_{10}x.$
Answer$\log_{10}3 + 1$
$= \log_{10} 3 + \log_{10} 10$
$= \log_{10} (3 \times 10)$
$= \log_{10}30.$
View full question & answer→Question 22 Marks
Solve for $x : \frac{\log 81}{\log 9}= x$
Answer$\frac{\log 81}{\log 9}=x$
$\Rightarrow \frac{\log 3^4}{\log 3^2}=x$
$\Rightarrow \frac{4 \log 3}{2 \log 3}=x$
$\Rightarrow x=2$
View full question & answer→Question 32 Marks
Solve for $x : \frac{\log 27}{\log 243}= x$
Answer$ \frac{\log 27}{\log 243}= x$
$\Rightarrow \frac{\log 3^3}{\log 3^5}= x$
$\Rightarrow \frac{3 \log 3}{5 \log 5}= x$
$\Rightarrow x=\frac{3}{5}$
View full question & answer→Question 42 Marks
Solve for $x: \log (x + 5) = 1$
Answer$\log (x + 5) = 1$
$= \log 10$
$\Rightarrow x + 5 = 10$
$\Rightarrow x$
$= 10 - 5$
$= 5.$
View full question & answer→Question 52 Marks
If $a b + b \log a - 1 = 0,$ then prove that $b^a.a^b = 10$
AnswerGiven $a \log b + b \log a - 1 = 0$
$\Rightarrow \log(b)^a + \log(a)^b- \log 10 = 0$
$\Rightarrow \log(b^a . a^b) = \log 10$
$\therefore b^a . a^b = 10.$
View full question & answer→Question 62 Marks
Express the following in a form free from logarithm$:2 \log x+\frac{1}{2} \log y=1$
Answer$2 \log x+\frac{1}{2} \log y=1$
$\Rightarrow \log x^2+\log \sqrt{y}=\log 10$
$\Rightarrow \log \left(x^2 \sqrt{y}\right)=\log 10$
$\Rightarrow x^2 \sqrt{y}=10$
View full question & answer→Question 72 Marks
Express the following in a form free from logarithm$:2 \log x + 3 \log y = \log a$
Answer$2 \log x + 3 \log y = \log a$
$\Rightarrow \log x^2 + \log y^3 = \log a$
$\Rightarrow \log (x^2. y^2) = \log a$
$\Rightarrow x^2y^3 = a.$
View full question & answer→Question 82 Marks
Simplify $: \log b \div \log b^2$
Answer$\log b \div \log b^2$
$=\log b \div 2 \log b$
$=\frac{\log b}{2 \log b}$
$=\frac{1}{2} .$
View full question & answer→Question 92 Marks
Simplify$: \log a^2 + \log a^{-1}$
Answer$\log a^2 + \log a^{-1}$
$= 2 \log a + (-1) \log a$
$= 2 \log a - \log a$
$= \log a (2 - 1)$
$= \log a.$
View full question & answer→Question 102 Marks
Express the following in terms of $\log 5$ and$/$or $\log 2: \log 250$
Answer$\log 250$
$= \log (5^3 \times 2)$
$= \log 5^3 + \log 2$
$= 3 \log 5 + \log 2.$
View full question & answer→Question 112 Marks
Express the following in terms of $\log 5$ and$/$or $\log 2: \log 500$
Answer$\log 500$
$= \log (2^2 \times 5^3)$
$= \log 2^2+ \log 5^3$
$= 2 \log 2 + 3 \log 5.$
View full question & answer→Question 122 Marks
Express the following in terms of $\log 5$ and$/$or $\log 2: \log 160$
Answer$\log 160$
$= \log (2^5 \times 5)$
$= \log 2^5 + \log 5$
$= 5 \log 2 + \log 5.$
View full question & answer→Question 132 Marks
If $\log 4 = 0.6020,$ find the value of each of the following $:\log 2.5$
Answer$\log 2.5$
$=\log \left(\frac{10}{4}\right) $
$ =\log 10-\log 4$
$ =1-0.6020$
$=0.3980 .$
View full question & answer→Question 142 Marks
If $\log 4 = 0.6020,$ find the value of each of the following $:\log 8$
Answer$\log 8$
$\log 4=0.6020$
$\Rightarrow \log 2^2=0.6020$
$\Rightarrow 2 \log 2=0.6020$
$\Rightarrow \log 2=\frac{0.6020}{2}$
$=0.3010$
View full question & answer→Question 152 Marks
Express the following in terms of $\log 5$ and/or $\log 2: \log 80$
Answer$\log 80$
$= \log (2^4 \times 5)$
$= \log 2^4 + \log 5$
$= 4 \log 2 + \log 5.$
View full question & answer→Question 162 Marks
If $\log 2 = x$ and $\log 3 = y,$ find the value of each of the following on terms of $x$ and $y: \log60$
Answer$log60$
$= \log (2 \times 3 \times 10)$
$= \log 2 + \log 3 + \log 10$
$= x + y + 1.$
View full question & answer→Question 172 Marks
Express the following in terms of $\log 5$ and/or $\log 2: \log 20$
Answer$\log 20$
$= \log (2^2 \times 5)$
$= \log 2^2 + \log 5$
$= 2 \log 2 + \log 5.$
View full question & answer→Question 182 Marks
If $\log_{10}25 = x$ and $\log_{10}27 = y;$ evaluate without using logarithmic tables, in terms of $x$ and $y: \log_{10}3$
Answer$\log _{10} 27=y$
$\Rightarrow \log _{10} 3^3=y$
$\Rightarrow 3 \log _{10} 3=y$
$\Rightarrow \log _{10} 3=\frac{y}{3} .$
View full question & answer→Question 192 Marks
If $\log _{10} 25=x$ and $\log _{10} 27=y ;$ evaluate without using logarithmic tables, in terms of $x$ and $y: \log _{10} 5$
Answer$\log _{10} 25=x$
$\Rightarrow \log _{10} 5^2= x$
$\Rightarrow 2 \log _{10} 5=x$
$\Rightarrow \log _{10} 5=\frac{x}{2} .$
View full question & answer→Question 202 Marks
If $2 \log x + 1 = 40,$ find$: x$
Answer$2 \log x+1=40$
$\Rightarrow 2 \log x=39$
$\Rightarrow \log x^2=39$
$\Rightarrow x^2=10^{39}$
$\Rightarrow x=10^{\frac{39}{2}}$
View full question & answer→Question 212 Marks
If $\log x = a$ and $\log y = b,$ write down $10^{a-1}$ in terms of $x$
Answer$10^{ a -1}$ in terms of $x$
$\log x = a$
$\Rightarrow x =10^{ a }$
$\therefore 10^{ a -1}$
$=\frac{10^{ a }}{10}$
$=\frac{x}{10} .$
View full question & answer→Question 222 Marks
Express the following in terms of $\log 2$ and $\log 3: \log 648$
Answer$\log 648$
$= \log (2^3 \times 3^4)$
$= \log 2^3+ 3^4$
$= 3 \log 2 + 4 \log 3.$
View full question & answer→Question 232 Marks
Express the following in terms of $\log 2$ and $\log 3: \log 216$
Answer$\log 216$
$= \log (2^3\times 3^3)$
$= \log 2^3 + \log 3^3$
$= 3 \log 2 + 3 \log 3.$
View full question & answer→Question 242 Marks
If $\log a=p$ and $\log b=q,$ express $\frac{a^3}{b^2}$ in terms of $p$ and $q$.
Answer$\log \frac{a^3}{b^2}$
$=\log a^3-\log b^2$
$=3 \log a-2 \log b$
$=3 p-2 q .$
View full question & answer→Question 252 Marks
Express the following in terms of $\log 2$ and $\log 3: \log 144$
Answer$\log 144$
$= \log (2^4 \times 3^2)$
$= \log 2^4 + \log 3^2$
$= 4 \log 2 + 2 \log 3.$
View full question & answer→Question 262 Marks
If $\log _{10}a = x, \log_{10} b = y$ and $\log _{10} c = z,$ find $10^{3y -1}$ in terms of $b$
Answer$10^{3 y-1}$ in terms of $b$
$\log _{10} b=y$
$\Rightarrow b=10^y$
$10^{3 y-1}$
$=(10 y)^3 \cdot 10^{-1}$
$=\frac{b^3}{10} .$
View full question & answer→Question 272 Marks
If $\log _{10}a = x, \log_{10} b = y$ and $\log _{10} c = z,$ find $10^{2x-3}$ in term of $a$
Answer$10^{2 x-3}$ in term of $a$
$\log _{10} a=x$
$\Rightarrow a=10^x$
$10^{2 x-3}$
$=\left(10^x\right)^2 \cdot 10^{-3}$
$=\frac{a^2}{1000} .$
View full question & answer→Question 282 Marks
If $\log _{10} x = p,$ express the following in terms of $x: 10^{2-p}$
Answer$10^{2-p}$
$\log _{10} x=p$
$\Rightarrow x=10^p$
$\therefore 10^{2-p}$
$=10^2 \cdot 10^{-p}$
$=100 \cdot x^{-1}$
$=\frac{100}{x} .$
View full question & answer→Question 292 Marks
If $\log _{10} x = p,$ express the following in terms of $x: 10^{2p-3}$
Answer$10^{2 p-3}$
$\log _{10} x=p$
$\Rightarrow x=10^p$
$\therefore 10^{2 p-3}$
$=10^{2 p} \cdot 10^{-3}$
$=\left(10^p\right)^2 \cdot 10^{-3}$
$=\frac{x^2}{1000} .$
View full question & answer→Question 302 Marks
If $\log _{10}x = p,$ express the following in terms of $x: 10^{p+1}$
Answer$10^{p+1}$
$\log _{10}x = p$
$\Rightarrow x = 10^p$
$\therefore 10^{p+1}$
$= 10^p . 10^1$
$= 10x.$
View full question & answer→Question 312 Marks
If $\log _{10} m = n,$ express the following in terms of $m: 10 ^{-3n}$
Answer$10^{-3 n}$
$\log _{10} m = n$
$\Rightarrow m =10^{ n }$
$\therefore 10^{-3 n }$
$=\left(10^{ n }\right)^{-3}$
$=( m )^{-3}$
$=\frac{1}{ m ^3} .$
View full question & answer→Question 322 Marks
If $\log _{10} m = n,$ express the following in terms of $m: 10 ^{2n+1}$
Answer$10 ^{2n+1}$
$\log _{10} m = n$
$\Rightarrow m = 10 ^n$
$\therefore 10 ^{2n+1}$
$= 10 ^{2n} . 10^1$
$= (10^n)^2 . 10$
$= 10 m^2.$
View full question & answer→Question 332 Marks
If $\log _{10} m = n,$ express the following in terms of $m: 10^{n -1}$
Answer$10^{n-1}$
$\log _{10} m=n$
$\Rightarrow m=10^n$
$\therefore 10^{n-1}$
$=10^n \cdot 10^{-1}$
$=\frac{m}{10} .$
View full question & answer→Question 342 Marks
If $\log _{10} x = a,$ express the following in terms of $x: 10^{2a -3}$
Answer$10^{2 a-3}$
$\log _{10} x = a$
$\Rightarrow x =10^{ a }$
$\therefore 10^{2 a -3}$
$=10^{2 a } \cdot 10^{-3}$
$=\left(10^{ a }\right)^2 10^{-3}$
$=\frac{x^2}{1000} .$
View full question & answer→Question 352 Marks
If $\log _{10} x = a,$ express the following in terms of $x: 10 ^{-a}$
Answer$10^{-a}$
$\log _{10} x = a$
$\Rightarrow x =10^{ a }$
$\therefore 10^{- a }$
$= x ^{-1}$
$=\frac{1}{x} .$
View full question & answer→Question 362 Marks
If $\log _{10} x = a,$ express the following in terms of $x : 10 ^{2a}$
Answer$10 ^{2a}$
$\log _{10}x = a$
$\Rightarrow x = 10 ^a$
$\therefore 10 ^{2a} = (10^a)^2$
$= x^2.$
View full question & answer→Question 372 Marks
Find the value of: $\log _{0.1} 10$
Answer$\log _{0.1} 10$
Let $\log _{0.1} 10 = x$
$\Rightarrow 0.1^x = 10$
$\Rightarrow (10^{-1})^x = 10^1$
$\Rightarrow x = -1$
$\therefore -x = 1.$
View full question & answer→Question 382 Marks
Find the value of: $\log _a a^3$
Answer$\log _aa^3$
Let $\log _aa^3 = x$
$\Rightarrow a^x = a^3$
$\therefore x = 3.$
View full question & answer→Question 392 Marks
Find the value of: $\log _2 8$
Answer$\log _2 8$
Let $\log _2 8 = x$
$\Rightarrow 2^x= 8$
$\Rightarrow 2^x = 2^3$
$\therefore x = 3.$
View full question & answer→Question 402 Marks
Find the value of: $\log _5 \frac{1}{25}$
Answer$\log _5 \frac{1}{25} $
Let $\log _5 \frac{1}{25}= x $
$ \Rightarrow 5^{ x }=\frac{1}{25} $
$\Rightarrow 5^{ x }=5^{-2}$
$ \therefore x =-2 .$
View full question & answer→Question 412 Marks
Find the value of: $\log _3 81$
Answer$\log _3 81$
Let $\log _3 81 = x$
$\Rightarrow 3^x = 81$
$\Rightarrow 3^x = 3^4$
$\therefore x = 4.$
View full question & answer→Question 422 Marks
Find the value of: $\log _{\frac{1}{2}} 16$
Answer$\log _{\frac{1}{2}} 16$
Let $\log _{\frac{1}{2}} 16= x$
$\Rightarrow 16=\left(\frac{1}{2}\right)^x $
$ \Rightarrow 2^{-x}=2^4 $
$ \Rightarrow x=-4 $
$\therefore-x=4 .$
View full question & answer→Question 432 Marks
Find the value of: $\log _5 125$
Answer$\log _5 125$
Let $\log _5 125 = x$
$\Rightarrow 125 = 5^x$
$\Rightarrow 5^x = 5^3$
$\therefore x = 3.$
View full question & answer→Question 442 Marks
Find the value of: $\log _{10} 0.0001$
Answer$\log _{10} 0.0001$
Let $\log _{10} 0.0001 = x$
$\Rightarrow 0.0001 = 10^x$
$\Rightarrow 10^x = 10^{-4}$
$\therefore x = -4.$
View full question & answer→Question 452 Marks
Find the value of: $\log _{\frac{1}{5}} 125=x$
Answer$\log _{\frac{1}{5}} 125 $
Let $\log _{\frac{1}{5}} 125= x $
$ \Rightarrow\left(\frac{1}{5}\right)^x=125 $
$\Rightarrow 5^{- x }=5^3 $
$ \therefore- x =3 $
$\Rightarrow x =-3 .$
View full question & answer→Question 462 Marks
Find the value of: $\log _2128$
Answer$\log _2128$
Let $\log _2 125 = x$
$\Rightarrow 2^x = 128$
$\Rightarrow 2^x= 2^7$
$\therefore x = 7.$
View full question & answer→Question 472 Marks
Find the value of: $\log _5 3125$
Answer$\log _5 3125$
Let $\log _5 3125$
$\Rightarrow 5^x = 3125$
$\Rightarrow 5^x = 5^5$
$\therefore x = 5.$
View full question & answer→Question 482 Marks
Find the value of: $\log _3 81$
Answer$\log _3 81$
Let $\log _381 = x$
$\Rightarrow 3^x= 81$
$\Rightarrow 3^x = 3^4$
$\therefore x = 4.$
View full question & answer→Question 492 Marks
Find the value of: $\log _{10} 1000$
Answer$\log_{10} 1000$
Let $\log_{10}1000 = x$
$\Rightarrow 10^x = 1000$
$\Rightarrow 10^x = 10^3$
$\therefore x = 3.$
View full question & answer→Question 502 Marks
Find $x$ in the following when$: \log _4 0.0625 = x$
Answer$\log _4 0.0625 = x$
$\Rightarrow 0.0625 = 4^x$
$\Rightarrow 4^x = 4^{-2}$
$\therefore x = -2.$
View full question & answer→