Question

1)log ( ) log ( ) - log ( ) log₂₅125 2)log ( )log₃₄₃ 49 + log₄ log₃ 3) log²₂ log () 49 * log³ ( ) log₂ 32

Answer 1

Решение задания смотри на фотографии

Answer 2

To simplify these logarithmic expressions, let's first recall some logarithmic properties:

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

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

1. log( ) log( ) - log( ) log₂₅125

To simplify this expression, we will use properties 1 and 2:

logₐ(b) - logₐ(c) = logₐ(b/c)

log( ) log( ) - log( ) log₂₅125 = log( ) - log(125) log₂₅

Now, we need to apply a logarithmic base conversion using the change of base formula:

logₐ(b) = log(c) / log(a)

log( ) = log( ) / log( )

log(125) = log(125) / log(25)

Therefore, the expression becomes:

log( ) - log(125) / log( ) log(25)

2. log( )log₃₄₃49 + log₄log₃

To simplify this expression, we will use properties 1 and 2:

logₐ(b) + logₐ(c) = logₐ(bc)

log( )log₃₄₃49 + log₄log₃ = log(49 * 343) log(4 * 3)

Now, calculate the values inside the logarithms:

49 * 343 = 16807 4 * 3 = 12

The expression becomes:

log(16807) log(12)

3. log²₂log()49 * log³()log₂32

To simplify this expression, we will use property 3:

logₐ(b^m) = m * logₐ(b)

log²₂log()49 * log³()log₂32 = 2 * log(49) * 3 * log(32)

Now, calculate the values inside the logarithms:

log(49) = log₇(7) = 1 (because 7^1 = 7) log(32) = log₂(32) = 5 (because 2^5 = 32)

The expression becomes:

2 * 1 * 3 * 5 = 30

So, the simplified expressions are:

1. log( ) - log(125) / log( ) log(25)
2. log(16807) - log(12)
3. 30