What is 4(ln(5)-ln(x))+(4ln(x)-ln(25)) ?

The solution to 4(ln(5)-ln(x))+(4ln(x)-ln(25)) is 2ln(5)

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simplify 4(ln(5)-ln(x))+(4ln(x)-ln(25))

simplify

4 (ln (5) - ln (x)) + (4 ln (x) - ln (25))

2 ln (5)

Decimal Notation

3.21887 \dots

Solution steps

4 (ln (5) - ln (x)) + (4 ln (x) - ln (25))

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (5) - ln (x) = ln (\frac{5}{x})

= 4 ln (\frac{5}{x}) + + 4 ln (x) - ln (25)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

4 ln (\frac{5}{x}) = ln ((\frac{5}{x})^{4})

= ln ((\frac{5}{x})^{4}) + + 4 ln (x) - ln (25)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

4 ln (x) = ln (x^{4})

= ln ((\frac{5}{x})^{4}) + + ln (x^{4}) - ln (25)

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

ln ((\frac{5}{x})^{4}) + ln (x^{4}) = ln ((\frac{5}{x})^{4} x^{4})

= ln ((\frac{5}{x})^{4} x^{4}) - ln (25)

(\frac{5}{x})^{4} x^{4} = 625

= ln (625) - ln (25)

ln (625) = 4 ln (5)

ln (25) = 2 ln (5)

= 4 ln (5) - 2 ln (5)

Add similar elements:

4 ln (5) - 2 ln (5) = 2 ln (5)

= 2 ln (5)

lo g_{10} (\frac{3 x ^{3}}{y})

3 lo g_{10} (8) + lo g_{10} (9) - lo g_{10} (36)

lo g_{3} (x^{3} y^{4})

lo g_{10} (6) (k^{2} - 36) - lo g_{10} (6 k) - 6

lo g_{10} (z) - lo g_{10} (2 y)

What is 4(ln(5)-ln(x))+(4ln(x)-ln(25)) ?
The solution to 4(ln(5)-ln(x))+(4ln(x)-ln(25)) is 2ln(5)