Question

2Log5 2/5+log5√9-log5 72/125+log5√10

Answer 1

I really do not understand this

Answer 2

To simplify the expression, we can use the properties of logarithms:

1. log(a) + log(b) = log(a * b)
2. log(a) - log(b) = log(a / b)
3. log(x^n) = n * log(x)

Using these properties, we can rewrite the expression:

2Log5 (2/5) + log5 (√9) - log5 (72/125) + log5 (√10)

= Log5 ((2/5)^2) + Log5 (√9) - Log5 (72/125) + Log5 (√10)

= Log5 (4/25) + Log5 (3) - Log5 (72/125) + Log5 (√10)

= Log5 ((4/25) * 3) - Log5 (72/125) + Log5 (√10)

= Log5 (12/25) - Log5 (72/125) + Log5 (√10)

Now, let's simplify the fractions inside the logarithms:

12/25 = (2^2 * 3) / (5^2) = (2^2 * 3) / (5^2 * 5^2) = (2^2 * 3) / 5^4 = 4 * 3/625 = 12/625

72/125 = (2^3 * 3^2) / (5^3) = (2^3 * 3^2) / (5^3 * 5^2) = (2^3 * 3^2) / 5^5 = 8 * 9/3125 = 72/3125

Now, substitute these simplified fractions back into the expression:

= Log5 (12/625) - Log5 (72/3125) + Log5 (√10)

Since 625 = 5^4 and 3125 = 5^5, we can simplify further:

= Log5 (12/5^4) - Log5 (72/5^5) + Log5 (√10)

Now, use the property log(x^n) = n * log(x) to move the exponent down:

= Log5 (12) - 4 * Log5 (5) - Log5 (72) + 5 * Log5 (5) + Log5 (√10)

Since log5 (5) = 1, we can simplify further:

= Log5 (12) - 4 + Log5 (72) + 5 + Log5 (√10)

Now, we simplify the square root of 10:

√10 = 10^(1/2) = 5^(1/2 * log5 10) = 5^(log5√10)

Therefore, Log5 (√10) = 1/2 * Log5 (10)

= 1/2 * 1 = 1/2

Now substitute all the simplified values back into the expression:

= Log5 (12) - 4 + Log5 (72) + 5 + 1/2

= Log5 (12 * 72) + 1/2

= Log5 (864) + 1/2

So, the simplified expression is Log5 (864) + 1/2.

Answer 3

I apologize for the confusion. Let's go through the simplification process step by step.

The given expression is:

2Log5 (2/5) + log5 (√9) - log5 (72/125) + log5 (√10)

First, let's work on the logarithms individually:

1. Log5 (2/5):
This can be simplified as (2/5)^(log5 2) using the logarithm base change property. However, let's first convert 2/5 to a power of 5:

2/5 = (2 * 5^0) / (5^1) = (2 * 5^0) / (5 * 5^0) = 2/5^1

Now, we can rewrite Log5 (2/5) as (2/5^(1))^log5 2 = (2/5^1)^log5 2 = (2/5)^log5 2

2. Log5 (√9):
The square root of 9 can be written as 9^(1/2). Therefore, Log5 (√9) = Log5 (9^(1/2)) = (9^(1/2))^log5 9 = 3^log5 9

3. Log5 (72/125):
Similar to the first term, let's rewrite 72/125 as a power of 5:

72/125 = (8 * 5^2) / (5^3) = (8/5 * 5^2) / (5 * 5^2) = 8/5^1 / 5^3

Now, we can rewrite Log5 (72/125) as (8/5^1 / 5^3)^log5 9 = (8/5)^log5 9 / 5^(1 * log5 9) = (8/5)^log5 9 / 5^log5 9

4. Log5 (√10):
We can rewrite this as Log5 (10^(1/2)) = (10^(1/2))^log5 10 = 10^(1/2 * log5 10) = 10^(log5√10)

Now that we have simplified the individual logarithms, let's substitute the values back into the original expression:

2Log5 (2/5) + log5 (√9) - log5 (72/125) + log5 (√10)

= 2[(2/5)^log5 2] + (3^log5 9) - [(8/5)^log5 9 / 5^log5 9] + 10^(log5√10)

Unfortunately, we cannot simplify this expression any further without knowing the specific values of log5 2, log5 9, and log5 10.

Please let me know if you have any additional questions or need further clarification.