Question

Log(2)1/3 + log(4)9 log(25)9 - log(5)3 Объясните пожалуйста более доступно, сегодня тему проходили, но я не поняла..

Answer 1

Log₂(1/3)+log₄9=log₂(1/3)+log₂²9=log₂(1/3)+log2√9=log₂(1/3)+log²3=
=log₂((1/3)*3)=log₂1=0.
log₂₅9-log₅3=log₅²9-log₅3=log₅√9-log₅3=log₅3-log₅3=0.

Answer 2

Logarithm Basics

Before explaining the specific expression you provided, let's start with some basics about logarithms.

A logarithm is a mathematical function that represents the exponent to which a base must be raised to obtain a given number. In other words, it tells us the power to which a base number must be raised to get the desired result.

The logarithm function is denoted as log(base)number. Here, the base can be any positive number greater than 1, and the number is the value for which we want to find the logarithm.

For example, if we have the expression log(2)8, it means we want to find the power to which 2 must be raised to obtain 8. In this case, the answer is 3 because 2^3 equals 8.

Now, let's apply this understanding to the expression you provided: log(2)1/3 + log(4)9 + log(25)9 - log(5)3.

Breaking Down the Expression

To simplify the expression, let's calculate each logarithm separately and then add or subtract the results.

1. log(2)1/3: This means we need to find the power to which 2 must be raised to obtain 1/3. To calculate this, we can rewrite 1/3 as a fraction with a power of 2 as the denominator. The result is log(2)(2^-1). Using the property of logarithms that log(base)(a^b) = b * log(base)(a), we can rewrite this as -1 * log(2)2. Since the logarithm of the base to itself is always 1, the expression simplifies to -1.

2. log(4)9: This means we need to find the power to which 4 must be raised to obtain 9. Rewriting 9 as a power of 4, we have log(4)(4^2). Using the property mentioned earlier, this becomes 2 * log(4)4. Again, the logarithm of the base to itself is always 1, so the expression simplifies to 2.

3. log(25)9: This means we need to find the power to which 25 must be raised to obtain 9. Rewriting 9 as a power of 25, we have log(25)(25^(2/3)). Using the property mentioned earlier, this becomes (2/3) * log(25)25. Once again, the logarithm of the base to itself is always 1, so the expression simplifies to 2/3.

4. log(5)3: This means we need to find the power to which 5 must be raised to obtain 3. There is no simple way to express this as an integer power, so we'll leave it as log(5)3.

Simplified Expression

Now, let's put all the simplified logarithms together:

log(2)1/3 + log(4)9 + log(25)9 - log(5)3 simplifies to -1 + 2 + 2/3 - log(5)3.

Please note that the last term, log(5)3, cannot be simplified further without a calculator or additional information.

I hope this explanation helps clarify the topic for you! Let me know if you have any further questions.