Why is (log3 400)/2 equivalent to log3 20? Can you show me how you can prove that step by step, without a calculator? Thank you so much!
is
log 400 = 2 log 20 ??
well
log a^x = a log x
so
2 log 20 = log 20^2
and lo and behold
20^2 = 400
Oh, thank you so much! I did not think of making 400 20^2.
To prove that (log3 400)/2 is equivalent to log3 20, we need to apply some basic logarithmic properties and simplify the expression step-by-step. Here's how we can do it:
Step 1: Let's start by expressing 400 and 20 in terms of powers of 3.
- 400 can be expressed as 3^2 * 4^2 = (3^2 * 2^2 * 2^2)
- 20 can be expressed as 3 * 2 * 2 = (3^1 * 2^2)
Step 2: Applying the logarithmic property log(a * b) = log(a) + log(b), we can rewrite the expressions:
- log3 (400) = log3 (3^2 * 2^2 * 2^2) = log3 (3^2) + log3 (2^2) + log3 (2^2)
- log3 (20) = log3 (3^1 * 2^2) = log3 (3^1) + log3 (2^2)
Step 3: Applying another logarithmic property log(a^b) = b * log(a), we can simplify further:
- For log3 (400):
- log3 (3^2) = 2 * log3 (3)
- log3 (2^2) + log3 (2^2) = 2 * log3 (2) + 2 * log3 (2) = 4 * log3 (2)
- For log3 (20):
- log3 (3^1) = 1 * log3 (3)
- log3 (2^2) = 2 * log3 (2)
Step 4: Now, let's substitute the simplified expressions back into the original equation:
- (log3 400)/2 = (2 * log3 3 + 4 * log3 2) / 2 = log3 3 + 2 * log3 2
- log3 20 = log3 3 + 2 * log3 2
Step 5: As we can see, (log3 400)/2 is indeed equivalent to log3 20, as they have the same simplified form.
Therefore, we have shown step-by-step that (log3 400)/2 is equivalent to log3 20 without using a calculator.