2Log5 2/5+log5√9-log5 72/125+log5√10
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.
I really do not understand this
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.
To simplify the expression, let's break it down step-by-step:
Step 1: Simplify the first term, 2Log5 (2/5):
The logarithmic property logb (a) = logb (c) - logb (d) can be used to simplify the expression.
2Log5 (2/5) = Log5 ((2/5)^2) = Log5 (4/25)
Step 2: Simplify the second term, log5 √9:
Since the square root of 9 is 3, we can simplify log5 √9 as log5(3).
Step 3: Simplify the third term, log5 72/125:
We can simplify log5 (72/125) using the logarithmic property:
log5 (72/125) = log5(72) - log5(125)
Now, let's further simplify each term separately:
log5(72) = log5(8*9) = log5(8) + log5(9) = 3 + log5(9)
log5(125) = log5(5^3) = 3
Therefore,
log5 (72/125) = 3 + log5(9) - 3 = log5(9)
Step 4: Simplify the fourth term, log5 √10:
We can simplify log5 √10 according to the property logb(b) = 1 and logb (a) + logb (c) = logb (a * c):
log5 √10 = 1/2 * log5 (10) = 1/2 * log5(2 * 5) = 1/2 * (log5(2) + log5(5)) = 1/2 * (log5(2) + 1)
Now, let's put all the simplified terms together:
2Log5 (2/5) + log5 √9 - log5(72/125) + log5 √10
= Log5 (4/25) + log5(3) - log5(9) + 1/2 * (log5(2) + 1)
This is the simplified expression.
To simplify the expression 2Log5 2/5 + log5√9 - log5 72/125 + log5√10, we need to apply the properties of logarithms.
First, let's simplify each individual term:
1. 2Log5 2/5:
We can use the power rule of logarithms, which states that Logbx^a = aLogbx.
Therefore, 2Log5 2/5 can be simplified as Log5 (2/5)^2.
Since (2/5)^2 = 4/25, the term becomes Log5 4/25.
2. log5√9:
We can use the property of logarithms that states Logba = c if and only if a^c = b.
Thus, log5√9 can be rewritten as 5^x = √9, where x is the unknown exponent.
The square root of 9 is 3, so 5^x = 3.
To solve for x, let's take the logarithm of both sides: Log5 (5^x) = Log5 3.
By the power rule of logarithms, xLog5 5 = Log5 3.
Since Log5 5 = 1 (logarithm base equals its argument), the equation simplifies to x = Log5 3.
Therefore, log5√9 can be replaced by Log5 3.
3. log5 72/125:
Similar to the previous step, we can use the property of logarithms to rewrite it as 5^x = 72/125.
To simplify the fraction, notice that 72 = 2^3 * 3^2, and 125 = 5^3.
Therefore, we can rewrite the equation as 5^x = (2^3 * 3^2) / 5^3.
Applying the property of dividing exponents with the same base, this can be further simplified to 5^x = 2^3 * 3^2 / 5^3 = (2^3 * 3^2) / (5^3 * 1).
Now, we can use the quotient rule of logarithms, which states that Logb(a / c) = Logb(a) - Logb(c).
Applying this rule, log5 72/125 becomes Log5 (2^3 * 3^2) - Log5 (5^3 * 1).
By expanding the logarithms, this simplifies to 3Log5 2 + 2Log5 3 - 3Log5 5.
4. log5√10:
Following similar steps as before, we express it as 5^x = √10.
The square root of 10 is not a perfect square or a multiple of any perfect square, so we leave it as Log5 √10.
Now, let's substitute these simplified terms back into the original expression:
Log5 4/25 + Log5 3 - (3Log5 5) + Log5 √10
Since we are adding and subtracting logarithms with the same base, we can apply the product and quotient rules of logarithms to combine them:
Log5 ((4/25) * 3 * (√10) / 5^3)
Now we can simplify the expression further using basic arithmetic operations:
Log5 ((12√10) / 125)
Finally, we have simplified the given expression to Log5 ((12√10) / 125).