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Multiple choice physics logarithms introduction to logarithm logarithmic notation exponential and logarithms

State whether true or false:
$\displaystyle \log F = \log : G + \log : m _1 + \log : m _2 - 2 \log : d$ gives $\displaystyle F = G \frac {m _2m _1}{d^2}$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$\displaystyle \log F = \log  \: G + \log  \: m _1 + \log  \: m _2 - 2 \: \log  \: d$

$\therefore \log F= \log G+\log m _1+\log m _2-\log d^2$....($\log a^b=b\log a$)

$\therefore \log F= \log \cfrac{Gm _1m _2}{d^2}$.....($\log a.b=\log a+\log b, \log \cfrac{a}{b}=\log a-\log b$)

$\therefore F=\cfrac{Gm _1m _2}{d^2}$
Multiple choice physics logarithms introduction to logarithm logarithmic notation exponential and logarithms

State whether true or false:
$\displaystyle \log : V = 2 \log : 2 - \log : 3 + \log : \pi + 3 \log : r$ gives $\displaystyle V = \frac {4}{3} \pi r^3$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\displaystyle \log : V = 2 : \log : 2 - : \log : 3 + : \log : \pi + 3 : \log : r$
$\therefore \log V = \log 2^2-\log 3+\log \pi+ \log r$....($\log a^b=b \log a$)
$\therefore \log V= \log \cfrac{4}{3}\pi r^3$.....($\log a.b = \log a + \log b, \log \cfrac {a}{b} = \log a - \log b$)
$\therefore V=\cfrac{4}{3}\pi r^3$

Multiple choice physics logarithms introduction to logarithm logarithmic notation exponential and logarithms

$\log V = 2 \log 2 - \log 3 + \log \pi + 3 \log r$ can be expressed as

  1. $V = \dfrac{4}{3} \pi r^{3}$
  2. $ V = \dfrac{2}{3} \pi r^{3}$
  3. $ V = \dfrac{4}{3} \pi r$
  4. $ V = \dfrac{2}{3} \pi r$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$ \log { V }  = 2\log { 2 } -\log { 3 } +\log { \pi  }  + 3\log { r }$
$ \log V = \log ({ 2 }^{ 2 }\times \pi \times { r }^{ 3 }) - \log { 3 }$
$ \log V = \log \dfrac { 4\pi { r }^{ 3 } }{ 3 } $
Removing log
$V = \dfrac {4}{3} \pi r^3$
Multiple choice physics logarithms introduction to logarithm logarithmic notation exponential and logarithms

Which of the following is true for $\log _25$?

  1. An integer

  2. A rational number

  3. An irrational number

  4. A whole number
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let us assume that $\log _{ 2 }{ 5 } =\frac { p }{ q } $ , where $p,q$ are integers

We have $5=2^{\frac{p}{q}}$
$\Rightarrow 5^{q}=2^{p}$
This suggest that $5$ and $2$ are not mutually prime , But $2$ and $5$ are mutually prime
Therefore $\log _{ 2 }{ 5 } $ is an irrational number