Is log 3 base 10 is rational or irrational?

The Fundamental Theorem of Arithmetic is that every integer is a product of primes.

How do you prove that a log of 3 is irrational?

log(3) = x/y where x and y are integers. Since 3 raised to any integer power is odd and 10 raised to any integer power is even and that a number cannot be both even and odd, this cannot be true! Hence, we have reached at a contradiction and so log(3) must be irrational.

How do you prove that log 5 to the base 3 is irrational?

Let us assume that is rational, then it can be written in the form , where p and q are both positive integers. Therefore, 3 must divide 5, but both 3 and 5 are co prime and 3 cannot divide 5, hence our assumption was wrong. Therefore, is irrational.

Is log to the base 10 irrational?

Both 10p and 2q are integers, by construction. But 10p is divisible by 5, while 2p, which has only 2 as a prime factor, is not. So 10p≠2q. So, by Proof by Contradiction, it follows that log102 is irrational.

Is log10 rational or irrational?

log10 5 is an irrational number. justification : let log10 5 is a rational number then it is in the form of P/Q , where P and Q are positive integers.

Is log base 10 an irrational number?

In this definition, the base of the logarithm is 10. Therefore, it is referred to as decimal logarithmic function. It can not be concluded with certainty that decimal logarithmic function is rational or irrational. Now, let us discuss the fundamental theorem of arithmetic.

What is the value of log 3 base 10?

Value of Log 1 to 10 for Log Base 10

Common Logarithm to a Number (log10 x)	Log Value
Log 3	0.4771
Log 4	0.6020
Log 5	0.6989
Log 6	0.7781

Is log 10 base 5 rational or irrational?

log10 5 is an irrational number. so there is no integer multiple of 5 equal to 3 and vice – versa. so our assumption is wrong. therefore, log10 5 is not a rational number.it is an irrational number.

What is the value of log 3 base 3?

1
Logarithm base 3 of 3 is 1 .

Are log values rational or irrational?

In this short note we prove that logarithms of most integers are irrational. Theorem 1: The natural logarithm of every integer n ≥ 2 is an irrational number. Remark 1: It follows from the proof that for any base b which is a transcendental number the logarithm logb n of every integer n ≥ 2 is an irrational number.

Is log 2 is rational or irrational?

Hence, log 2 is irrational.

Are logarithms of most integers irrational?

In this short note we prove that logarithms of most integers are irrational. Theorem 1: The natural logarithm of every integer n is an irrational number.

Is the logarithm of $a^b$ rational?

$\\begingroup$It looks as if they may be talking about proving certain values oflogarithm functions are irrational, but I’m not sure what it would mean to say that the log function itself is irrational. It does not make sense to say “$b$ is irrational; therefore $a^b$ is rational.

Is $a^b$ rational or irrational?

It does not make sense to say “$b$ is irrational; therefore $a^b$ is rational. No matter what positive number $a$ is, as long as it’s not $1$, there will be irrational numbers $b$ such that $a^b$ is rational, and other irrational numbers $b$ such that $a^b$ is irrational.

How do you prove that $9$ is a rational number?

One can write $$ 2^{2\\log_2 3} = \\left(2^{\\log_2 3}ight)^2 = 3^2 = 9, ag1 $$ so that is rational. But in doing that you don’t need to know anything at all about rational or irrational numbers until that final step where you observe that $9$ is rational.