Question

Log5 3 + Log 4 16 - log 5 15 / 2 - log 5 2/5

Answer 1

Log₅ 3 + Log₄ 16 - log₅ 15 / 2 - log₅ 2/5 = Log₅ 3 + Log₄ 16 -( log₅ 15 / 2 + log₅ 2/5)

= Log₅ 3 + Log₄ 4² -( log₅ 15 / 2 * 2/5) = Log₅ 3 +2 Log₄4 - log₅ 3=

= Log₅ 3 +2 - log₅ 3 = 2+ Log₅ 3 /3 = 2+ Log₅ 1 =2+0=2

Answer 2

To simplify the expression, we'll use the following logarithmic identities:

1. log_a(b) + log_a(c) = log_a(b * c)
2. log_a(b) - log_a(c) = log_a(b / c)
3. log_a(b^n) = n * log_a(b)

Let's apply these identities to the given expression step by step:

1. log5(3) + log4(16) - log5(15) / 2 - log5(2/5)

Using identity 3, we can rewrite log4(16) as 2 * log4(4), as 16 = 4^2.

2. log5(3) + 2 * log4(4) - log5(15) / 2 - log5(2/5)

Using identity 3 again, we can rewrite log4(4) as 1 * log4(2^2), as 4 = 2^2.

3. log5(3) + 2 * (1 * log4(2^2)) - log5(15) / 2 - log5(2/5)

Now, let's simplify the logarithms:

4. log5(3) + 2 * log4(2) - log5(15) / 2 - log5(2/5)

Next, we'll evaluate the logarithms:

log4(2) ≈ 0.5 (approximately, as 4^0.5 = 2) log5(15) ≈ 1.18 (approximately, as 5^1.18 ≈ 15) log5(2/5) ≈ -0.43 (approximately, as 5^(-0.43) ≈ 2/5)

5. log5(3) + 2 * 0.5 - 1.18 / 2 - (-0.43)

Now, perform the arithmetic:

6. log5(3) + 1 - 0.59 + 0.43

Combine like terms:

7. log5(3) + 0.84

So, the simplified expression is log5(3) + 0.84.