What is 1/4 ln(10)+ln(5)-1/2 ln(7) ?

The solution to 1/4 ln(10)+ln(5)-1/2 ln(7) is ln((5*10^{1/4}sqrt(7))/7)

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simplify 1/4 ln(10)+ln(5)-1/2 ln(7)

simplify

\frac{1}{4} ln (10) + ln (5) - \frac{1}{2} ln (7)

ln (\frac{5 \cdot 1 0 ^{\frac{1}{4}} 7}{7})

Decimal Notation

1.21212 \dots

Solution steps

\frac{1}{4} ln (10) + ln (5) - \frac{1}{2} ln (7)

Apply log rule:

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

\frac{1}{4} ln (10) = ln (1 0^{\frac{1}{4}})

= ln (1 0^{\frac{1}{4}}) + ln (5) - \frac{1}{2} ln (7)

Apply log rule:

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

ln (1 0^{\frac{1}{4}}) + ln (5) = ln (5 \cdot 1 0^{\frac{1}{4}})

= ln (1 0^{\frac{1}{4}} \cdot 5) - \frac{1}{2} ln (7)

Apply log rule:

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

\frac{1}{2} ln (7) = ln (7^{\frac{1}{2}})

= ln (5 \cdot 1 0^{\frac{1}{4}}) - ln (7^{\frac{1}{2}})

Apply log rule:

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

ln (5 \cdot 1 0^{\frac{1}{4}}) - ln (7^{\frac{1}{2}}) = ln (\frac{5 \cdot 1 0 ^{\frac{1}{4}}}{7 ^{\frac{1}{2}}})

= ln (\frac{1 0 ^{\frac{1}{4}} \cdot 5}{7 ^{\frac{1}{2}}})

\frac{1 0 ^{\frac{1}{4}} \cdot 5}{7 ^{\frac{1}{2}}} = \frac{5 \cdot 1 0 ^{\frac{1}{4}} 7}{7}

= ln (\frac{5 \cdot 1 0 ^{\frac{1}{4}} 7}{7})

ln (\frac{a ^{- 2}}{b ^{1} c ^{- 3}})

lo g_{11} (\frac{x + y}{x ^{2} y})

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

\frac{1}{4} ln (2 + e^{3 x}) - \frac{5}{12} ln (2) - 15 \frac{x}{12} + c

- lo g_{10} (0.1 M)

What is 1/4 ln(10)+ln(5)-1/2 ln(7) ?
The solution to 1/4 ln(10)+ln(5)-1/2 ln(7) is ln((5*10^{1/4}sqrt(7))/7)