Question

how many digits are there in 4^16.5^25​

Answer 1

Hint

What decides the number of digits is the exponent of 10 because we use base 10.

For example, $10^{0}=1$, $10^{1}=10$, $10^{2}=100$, …. We see that the power of 10 decides the number of digits.

The number of digits is exactly one more than the exponent of 10. (characteristic of the logarithm)

Solution

$10^{x}=4^{16}\cdot 5^{25}$

$\implies 10^{x}=2^{32}\cdot 5^{25}$

$\implies 10^{x}=2^{32-25}\cdot 2^{25}\cdot 5^{25}$

$\implies 10^{x}=2^{7}\cdot 10^{25}$

Taking the logarithm of base 10 we find the exponent of 10.

$\implies x=\log(2^{7}\cdot 10^{25})$

$\implies x=25+7\log 2\approx 27.1$

We should be careful that $27$ is not the number of digits. Since $10^{0}=1$ we add $1$ to the logarithm. (This is called the characteristic of the logarithm.)

⇒ There are $\boxed{28}$ digits in $4^{16}\cdot 5^{25}$.