How come log5 25^4 reduces to 4 log5 25
then 4 times 2
then finally eight?
log x^n = n log x
so
log5 25^4 = 4 log5 25
25 = 5^2
so
4 log5 25 = 4 log5 5^2
= 4*2 log5 5
5^(log5 x) = x
5^(log5 5) = 5
so
log5 5 = 1
SO
4*2*1 = 8
To understand why log5(25^4) reduces to 4 log5(25), let's break down the steps involved:
1. Start with the expression log5(25^4). This expression is composed of two parts: 25^4 as the base and log5 as the logarithm with base 5.
2. The first step is to simplify the base, 25^4. Since 25 is equal to 5^2 (25 = 5^2), we can rewrite it as (5^2)^4. According to the exponentiation rule (a^m)^n = a^(m * n), we can simplify this further to 5^(2 * 4), which is equal to 5^8.
3. Now, going back to the original expression, log5(25^4), we have log5(5^8) since 25^4 simplifies to 5^8. According to the logarithmic property loga(a^b) = b, the base of the logarithm and the base of the exponent cancel out, leaving us with just the exponent 8.
4. Therefore, the expression log5(25^4) simplifies to log5(5^8), which is equal to 8.
Hence, the final result is 8, not 4 log5 25 or eight. It seems there may have been a mistake in the calculations mentioned in your question.