Question

1) 3log 25 +10log₀,₂₅16 -11log₀,₃ 2) loglog₄256 + log₅log₆ 3)log₂log₅ +loglog₃27

Answer 1

Решение задания смотри на фотографии

Answer 2

To simplify the given logarithmic expressions, let's use the following logarithmic properties:

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

Now, let's solve each expression step by step:

1. 3log₂₅ + 10log₀,₂₅₁₆ - 11log₀,₃

Step 1: Apply property 1 for the first two terms: 3log₂₅ + 10log₀,₂₅₁₆ = log₂₅(25³) + log₀,₂₅₁₆(₁₆¹⁰)

Step 2: Convert the bases of the logarithms to a common base, such as base 10: log₂₅(25³) = log₅(25³) * log₂(25) log₀,₂₅₁₆(₁₆¹⁰) = log₁₆(₁₆¹⁰) * log₀,₂₅(₁₆)

Step 3: Simplify the logarithms using known values: log₅(25³) = log₅(125) = 3 log₂(25) = log₂(5²) = 2 log₁₆(₁₆¹⁰) = 10 log₀,₂₅(₁₆) = log(₁₆) / log(₂₅) ≈ 4 / 3

Step 4: Substitute the simplified values back into the original expression: 3 + 10 * 10 - 11 * (4/3)

Step 5: Simplify the expression further: 3 + 100 - (44/3)

Step 6: Find a common denominator and combine the terms: (9 + 300 - 44) / 3 = 265 / 3

So, the simplified value of the expression is 265/3.

2. log(log₄(256)) + log₅(log₆(x))

Step 1: Simplify the innermost logarithm: log₄(256) = log₄(4^4) = 4

Step 2: Now the expression becomes: log(log₄(4)) + log₅(log₆(x))

Step 3: Simplify further: log(1) + log₅(log₆(x)) = 0 + log₅(log₆(x))

So, the simplified expression is log₅(log₆(x)).

3. log₂(log₅(x)) + log(log₃(27))

Step 1: Simplify the innermost logarithm: log₅(x) = log₅(5³) = 3

Step 2: Now the expression becomes: log₂(3) + log(log₃(27))

Step 3: Simplify further: log₂(3) + log(3) = log₂(3) + 1

No further simplification is possible for this expression.

So, the simplified expression is log₂(3) + 1.