What is (2ln(3)-4ln(5))/(7ln(5)-ln(3)) ?

The solution to (2ln(3)-4ln(5))/(7ln(5)-ln(3)) is (ln(9/625))/(ln(78125/3))

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simplify (2ln(3)-4ln(5))/(7ln(5)-ln(3))

simplify

\frac{2 ln ( 3 ) - 4 ln ( 5 )}{7 ln ( 5 ) - ln ( 3 )}

\frac{ln ( \frac{9}{625} )}{ln ( \frac{78125}{3} )}

Decimal Notation

- 0.41706 \dots

Solution steps

\frac{2 ln ( 3 ) - 4 ln ( 5 )}{7 ln ( 5 ) - ln ( 3 )}

Apply log rule:

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

2 ln (3) = ln (3^{2})

= \frac{ln ( 3 ^{2} ) - 4 ln ( 5 )}{7 ln ( 5 ) - ln ( 3 )}

Apply log rule:

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

4 ln (5) = ln (5^{4})

= \frac{ln ( 3 ^{2} ) - ln ( 5 ^{4} )}{7 ln ( 5 ) - ln ( 3 )}

Apply log rule:

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

ln (3^{2}) - ln (5^{4}) = ln (\frac{3 ^{2}}{5 ^{4}})

= \frac{ln ( \frac{3 ^{2}}{5 ^{4}} )}{7 ln ( 5 ) - ln ( 3 )}

\frac{3 ^{2}}{5 ^{4}} = \frac{9}{625}

= \frac{ln ( \frac{9}{625} )}{7 ln ( 5 ) - ln ( 3 )}

Apply log rule:

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

7 ln (5) = ln (5^{7})

= \frac{ln ( \frac{9}{625} )}{ln ( 5 ^{7} ) - ln ( 3 )}

Apply log rule:

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

ln (5^{7}) - ln (3) = ln (\frac{5 ^{7}}{3})

= \frac{ln ( \frac{9}{625} )}{ln ( \frac{5 ^{7}}{3} )}

5^{7} = 78125

= \frac{ln ( \frac{9}{625} )}{ln ( \frac{78125}{3} )}

lo g_{10} (\frac{x}{10 y ^{5}})

lo g_{4} (x + 1) + lo g_{4} (4 x - 3)

lo g_{10} (10 x^{100}) 5

2 ln (x) - \frac{1}{7} ln (y)

lo g_{b} (\frac{1}{10}) + lo g_{b} (50)

What is (2ln(3)-4ln(5))/(7ln(5)-ln(3)) ?
The solution to (2ln(3)-4ln(5))/(7ln(5)-ln(3)) is (ln(9/625))/(ln(78125/3))