Math 9th Ex 2.2 | New book
Math 9th Ex 2.2 new book will be completely solved. We will check how to convert logarithmic form into exponential form and vice versa. Lets start exercise 2.2 math class 9 new book. Its the 2nd post of the series math class 9 chapter 2 Punjab board. $$Logarithmatic\;Form\\\underset{\boldsymbol a}{Log\;x}\;=\;y\\Exponential\;form\\\boldsymbol a^y\;=\mathit\;x\\$$
Tip for the conversion from exponetial form to logarithmic form or from logarithmic form to exponential is not to change the base.
Q1 Express each of the following in Logarithmic form:
(i) $$\mathit{10}^{\mathbf3}\;=\;1000\\Log_{\mathit{10}}1000\;=\boldsymbol\;\mathbf3$$
(ii) $$2^{\mathbf8}\;=\;256\\Log_2\;256\;=\boldsymbol\;\mathbf8\boldsymbol\;\\$$
(iii) $$3^{\boldsymbol-\mathbf3}\;=\;\frac1{27}\\Log_3\;\frac1{27}\;=\boldsymbol\;\boldsymbol-\mathbf3\boldsymbol\;\\$$
(iv) $$20^{\mathbf2}\;=\;400\\Log_{20}\;400\;=\boldsymbol\;\mathbf2\boldsymbol\;\\$$
(v) $$16^\frac{\boldsymbol-\mathbf1}{\mathbf4}\;=\;\frac12\\Log_{16}\;\frac12\;=\boldsymbol\;\frac{\boldsymbol-\mathbf1}{\mathbf4}\boldsymbol\;\\$$
(vi) $$11^{\mathbf2}\;=\;121\\Log_{11}\;121\;=\boldsymbol\;\mathbf2\boldsymbol\;\\$$
(vii) $$q^{\mathbf r}\;=\;p\\Log_q\;p\;=\boldsymbol\;\boldsymbol r\boldsymbol\;\\$$
(viii) $${(32)}^\frac{\boldsymbol-\mathbf1}{\mathbf5}\;=\;\frac12\\Log_{32}\;\frac12\;=\boldsymbol\;\frac{\boldsymbol-\mathbf1}{\mathbf5}\\$$
Q2 Express each of the following in Exponential form:
(i) $$\underset{\mathbf5}{Log\;\mathit{125}}\;=\;3\\\mathbf5^3\;=\;\mathit{125}\\$$
(ii) $$\underset{\mathbf2}{Log\;16}\;=\;4\\\mathbf2^4\;=\;\mathit{16}\\$$
(iii) $$\underset{\mathbf{23}}{Log\;1}\;=\;0\\\mathbf{23}^0\;=\mathit\;\mathit1\\$$
Log of 1 will be 0, however the base may be, because any real number raised to the power 0 gives us 1.
(iv) $$\underset{\mathbf5}{Log\;\mathbf5\;}\;=\;1\\5^1\;=\mathit\;\mathit5\\$$
Log of any number with the same base will gives us 1.
(v) $$\underset{\mathbf2}{Log\;\frac18\;}\;=\;-3\\\mathbf2^{-3}\;=\mathit\;\frac{\mathit1}{\mathit8}\\$$
(vi) $$\underset{\mathbf9}{Log\;3\;}\;=\;\frac12\\\mathbf9^\frac12\;=\mathit\;\mathit3\\$$
(vii) $$\underset{\mathbf{10}}{Log\;100000\;}\;=\;5\\\mathbf{10}^5\;=\mathit\;\mathit{100000}\\$$
(viii) $$\underset{\mathbf4}{Log\;\frac1{16}\;}\;=\;-2\\\\\mathbf4^{-2}\;=\mathit\;\mathit{16}\\$$
Q3 Find the value of x in each of the following:
(i) $$\underset x{Log\;64\;}\;=\;3\\\\$$
$$x^3\;=\;64\\\\x^3\;=\;4^3\\\\So.\;x\;=4\\\\$$
(ii) $$\underset5{Log\;1}\;=\;x\\\\$$
$$5^x\;=\;1\\\\5^x\;=\;5^0\\\\x=0\\\\$$
(iii) $$\underset x{Log\;8}\;=\;1\\$$
$$x^1\;=\;8\\\\x\;=\;8\\$$
(iv) $$\underset{10}{Log\;x}\;=\;-3\\$$
$$10^{-3}\;=\;x\\\\\frac1{10^3}\;=\;x\\\\\frac1{1000}\;=\;x\\\\x\;=\;0.001\\$$
(v) $$\underset4{Log\;x}\;=\;\frac32\\$$
$$4^\frac32\;=\;x\\\\{(\;2^2)}^\frac32\;=\;x\\\\2^{2\times\frac32}\;=\;x\\\\2^3\;=\;x\\\\x\;=\;8\\\\$$
(vi) $$\underset2{Log\;1024}\;=\;x\\\\$$
$$2^x\;=\;1024\\\\2^x\;=\;2^{10}\\\\x\;=\;10\\$$
Logarithm ( word ) consists of two Greek words. One is LOGOS and the other is ARITHMOS. Arithmos means ratio or proportion. A Scottish Mathematician John Napier first used the word Logarithm.
How many times a number is going to be multiplied by itself to get another number can be found by Logarithm. In other words, if we want to find the power we will use Logarithm. In the next exercise we will talk about common logarithm and also differentiate between common log and natural log. we will study about characteristics and mantissa in detail with the help of basic rules and logarithmic table.
Best of luck for your bright future.
