What is (ln(10))/(2ln(2)-ln(5)) ?

The solution to (ln(10))/(2ln(2)-ln(5)) is (ln(10))/(ln(4/5))

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simplify (ln(10))/(2ln(2)-ln(5))

simplify

\frac{ln ( 10 )}{2 ln ( 2 ) - ln ( 5 )}

\frac{ln ( 10 )}{ln ( \frac{4}{5} )}

Decimal Notation

- 10.31885 \dots

Solution steps

\frac{ln ( 10 )}{2 ln ( 2 ) - ln ( 5 )}

Apply log rule:

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

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

= \frac{ln ( 10 )}{ln ( 2 ^{2} ) - ln ( 5 )}

Apply log rule:

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

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

= \frac{ln ( 10 )}{ln ( \frac{2 ^{2}}{5} )}

2^{2} = 4

= \frac{ln ( 10 )}{ln ( \frac{4}{5} )}

\frac{9}{10} ln (30)

- \frac{9}{20} \cdot lo g_{2} (\frac{9}{20}) - \frac{1}{20} \cdot lo g_{2} (\frac{1}{20})

\frac{3 ln ( 20 )}{2 ln ( 19 ) - ln ( 20 )}

- \frac{5}{14} \cdot lo g_{2} (\frac{5}{14}) - \frac{9}{14} \cdot lo g_{2} (\frac{9}{14})

What is (ln(10))/(2ln(2)-ln(5)) ?
The solution to (ln(10))/(2ln(2)-ln(5)) is (ln(10))/(ln(4/5))