Is $\log 2$ rational or irrational? Justify your answer.

Important Questions on Logarithm and its Applications

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

If $\log x=1.2500 \text{and} y=x^{\log x}$, then what is logy equal to?

HARD

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

Let $f$ be a function defined on the set of all positive integers such that $f\left(xy\right)=f\left(x\right)+f\left(y\right)$ for all positive integers $x, y.$ If $f\left(12\right)=24$ and $f\left(8\right)=15$ , the value of $f\left(48\right)$ is

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

What is ${\log }_{10}31.25$ equal to?

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

If ${\log }_{4}x+{\log }_{16}x+{\log }_{64}x+{\log }_{256}x=\frac{25}{6}$ then the value of $x$ is

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

If ${\log }_5{\log }_8\left(x^2-1\right)=0$, then a possible value of $\mathrm{x}$ is

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

What is the Value of ${\log }_{10}\left(\cos \mathrm{\theta }\right)+{\log }_{10}\left(\sin \mathrm{\theta }\right)+{\log }_{10}\left(\tan \mathrm{\theta }\right)+{\log }_{10}\left(\cot \mathrm{\theta }\right)+{\log }_{10}\left(\sec \mathrm{\theta }\right)+{\log }_{10}\left(\mathrm{cosec} \mathrm{\theta }\right)$?

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

If ${\log }_{10}x+{\log }_{10}{x}^2=2{\log }_{10}x+1$, then what is the value of $x$?

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

If $\underset{0}{\overset{\frac{\pi }{2 }}{\int }}\log \cos x dx=\frac{\pi }{2}\log \left(\frac{1}{2}\right),$ then $\underset{0}{\overset{\frac{\pi }{2 }}{\int }}\log \sec x dx=$

HARD

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

The value of $0.{04}^{{\log }_{\sqrt{5}}\left(\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots \right)}$ is

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

What is the solution of the equation $x{\log }_{10}\left(\frac{10}{3}\right)+{\log }_{10}3={\log }_{10}(2+3^x)+x$?

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

Consider the following statements:
1. ${\log }_{10} 50$ is a rational number.
2. ${\log }_{100} 10$ is an irrational number.
Which of the statememnts given above is/are correct? 

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

The inequality ${\log }_a\left\{f\right(x\left)\right\}<{\log }_a\left\{g\right(x\left)\right\}$ implies that

HARD

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

Let $S$ be the sum of the digits of the number ${15}^2\times 5^{18}$ in base $10$. Then,

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

There are $n$ zeros appearing immediately after the decimal point in the value of ${\left(0.2\right)}^{25}$. It is given that the value of ${\log }_{10}2=0.30103$. The value of $n$ is

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

If $S=\frac{1}{2}\left(\frac{1}{2}+\frac{1}{3}\right)-\frac{1}{4}\left(\frac{1}{2^2}+\frac{1}{3^2}\right)+\frac{1}{6}\left(\frac{1}{2^3}+\frac{1}{3^3}\right)\dots \dots \dots$ then $S=$

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

Find the value of ${\log }_5\sqrt{625}$

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

The value of ${\log }_{2}32=$

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

If $5^{x-1}={\left(2.5\right)}^{{{\log }_{10}}^5}$, then what is the value of $x$?

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

The inequality ${\log }_2\frac{3x-1}{2-x}<1$ holds true for

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

The value of ${\left({\log }_{3}30\right)}^{-1}+{\left({\log }_{4}900\right)}^{-1}+{\left({\log }_{5}30\right)}^{-1}$ is