We are given that log10 2<0.302. How many digits are there in the decimal representation of 5^500?

To determine the number of digits in the decimal representation of 5^500, we need to calculate the actual value of 5^500 first. Then, we can count the number of digits in that value.

Now, let's begin by using the logarithmic identity log(base a) b = c, which can be rewritten as b = a^c.

In this case, we have log10 2 < 0.302. Therefore, we can rewrite it as 2 < 10^(0.302).

Using the property that 10^(log a) = a, we can rewrite 10^(0.302) => 2 < 10^0.302.

Since the inequality holds true, we can say that 2 < 10^(0.302).

Next, let's calculate 5^500, which is the number we are interested in.

To evaluate this, we can use the property that (a^b)^c = a^(b*c).

Therefore, (5^2)^250 = 10^500.

Since 5^2 = 25, we can rewrite the above equation as 25^250 = 10^500.

Now, we know that 2 < 10^(0.302), therefore, we can conclude that 25^250 < 10^500.

Hence, 5^500 is less than 10^500.

Since 10^500 has 501 digits, we can deduce that the decimal representation of 5^500 will have 500 digits.

Thus, there are 500 digits in the decimal representation of 5^500.