9th Class Mathematics Related to Competitive Exam Antilogarithms

         Antilogarithms

The logarithm of a number always contains two parts which are characteristic and mantissa. The integral part is known as characteristic and the decimal part is known as mantissa. Mantissa is always kept positive. The number whose logarithm is \[x\] is called the antilogarithm of \[x\] and is denoted by antilog \[x\].  

The value of \[{{\log }_{343}}7\] is:

(a) 0                                                      

(b) 7

(c) \[\frac{1}{3}\]                            

(d) \[\frac{1}{7}\]

(e) None of these  

Answer: (c)  

Explanation:

let \[{{\log }_{343}}7=x,\] then \[{{343}^{x}}=7\]  

\[\Rightarrow \]\[{{({{7}^{3}})}^{x}}=7\]\[\Rightarrow \]\[{{7}^{3x}}=7\]\[\Rightarrow \]\[3x=1\]\[\Rightarrow \]\[x=\frac{1}{3}\]      

If \[\log 2+\frac{1}{2}\log x+\frac{1}{2}\log y=\log (x+y)\], then:

(a) \[x=y\]                                          

(b) \[x+y=1\]

(c) \[x=2y\]                                        

(d) \[x-y=1\]

(e) None of these  

Answer: (a)  

Explanation:

\[\log 2+\frac{1}{2}\log x+\frac{1}{2}\log y=\log (x+y)\]

\[\Rightarrow \]\[\log (2\times \sqrt{x}\times \sqrt{y})=\log (x+y)\]

\[\Rightarrow \]\[{{(x-y)}^{2}}=0\]                         

\[\Rightarrow \]    \[x=y\]