Log75+2log2-log3
useing the rules of logs, you have
log(75 * 2^2 / 3) = log(25*4) = log(100) = 2
Log 25
Well, well, well, looks like we've got ourselves a logarithmic party over here! Let me grab my logarhythmic beatbox. Ahem.
Alright, let's break it down. We have log75, which I like to call the "75 log-up." Then we have 2log2, which is the "double trouble log2." And to top it off, we have log3, where 3 is just tryin' to steal the logarithmic spotlight.
Now, let me put on my mathematical clown shoes and do some calculations.
log75 can be rewritten as log(25 * 3), which becomes log25 + log3. And since log25 is equal to 2, we have 2 + log3.
Next up, we have 2log2. Now, that's just 2 times log2, which is basically two times the "log of the furious."
Lastly, we have log3, which, well, is just log3. Let's not make it too complicated.
So, putting it all together, we have 2 + log3 + 2log2.
And that, my friend, is the delightful result of your logarithmic expression.
To simplify the expression log75+2log2-log3, we can use the logarithmic properties to combine the terms.
1. Apply the power rule of logarithms: log(x^a) = alog(x)
log75 = log(3 × 5^2)
2. Use the product rule of logarithms: log(ab) = log(a) + log(b)
log(3 × 5^2) = log(3) + log(5^2)
3. Simplify log(5^2) using the power rule: log(x^a) = alog(x)
log(5^2) = 2log(5)
Now, substituting the simplified forms back into the original expression:
log75 + 2log2 - log3 = log(3) + log(5^2) + 2log(2) - log(3)
Since log(3) and -log(3) cancel each other out, the expression simplifies to:
log75 + 2log2 - log3 = log(5^2) + 2log(2)
Finally, applying the power rule to log(5^2) and log(2):
log(5^2) + 2log(2) = 2log(5) + 2log(2)
Therefore, log75 + 2log2 - log3 simplifies to 2log(5) + 2log(2).
To solve the expression log75 + 2log2 - log3, we can use several logarithmic identities:
1. Logarithmic Identity: log a + log b = log (a * b)
2. Logarithmic Identity: log a - log b = log (a / b)
3. Logarithmic Identity: log a^n = n * log a
Let's break down the expression step by step:
1. log75 + 2log2 - log3
Since we don't have any common bases, we can't combine the terms directly. So, we need to simplify each term separately.
2. Simplifying log75:
We want to rewrite 75 as a power of a base, preferably a prime base. However, 75 doesn't have a prime base representation, but we can express it as a product of its prime factors: 75 = 3 * 5^2.
Using the logarithmic identity log a^n = n * log a, we can rewrite log (3 * 5^2) as log 3 + 2log 5.
So, now we have log75 = log3 + 2log5.
3. Simplifying 2log2:
Similarly, we can use the logarithmic identity log a^n = n * log a to simplify 2log2.
Since 2 is the base itself, we can rewrite it as 2log2 = log2^2.
So, now we have 2log2 = log (2^2) = log4.
4. Simplifying -log3:
There are no logarithmic identities that directly simplify this term, so we keep it as -log3.
5. Putting it all together:
Now, we can substitute the simplified terms back into the original expression:
log75 + 2log2 - log3 = (log3 + 2log5) + log4 - log3.
We can now combine the terms that have the same base using the logarithmic identity log a + log b = log (a * b).
So, (log3 + 2log5) + log4 - log3 = log (3 * 5^2) + log4 - log3.
Calculating further,
= log (3 * 25) + log4 - log3
= log 75 + log4 - log3
Now, we can combine the remaining terms using the logarithmic identity log a + log b = log (a * b) and log a - log b = log (a / b).
= log (75 * 4 / 3)
= log (300 / 3)
= log 100
= 2
Therefore, log75 + 2log2 - log3 equals 2.