What is 9/10 ln(10000)+1/10 ln(5000) ?

The solution to 9/10 ln(10000)+1/10 ln(5000) is ln(5000*2^{9/10})

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simplify 9/10 ln(10000)+1/10 ln(5000)

simplify

\frac{9}{10} ln (10000) + \frac{1}{10} ln (5000)

ln (5000 \cdot 2^{\frac{9}{10}})

Decimal Notation

9.14102 \dots

Solution steps

\frac{9}{10} ln (10000) + \frac{1}{10} ln (5000)

Apply log rule:

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

\frac{9}{10} ln (10000) = ln (1000 0^{\frac{9}{10}})

= ln (1000 0^{\frac{9}{10}}) + \frac{1}{10} ln (5000)

Apply log rule:

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

\frac{1}{10} ln (5000) = ln (500 0^{\frac{1}{10}})

= ln (1000 0^{\frac{9}{10}}) + ln (500 0^{\frac{1}{10}})

Apply log rule:

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

ln (1000 0^{\frac{9}{10}}) + ln (500 0^{\frac{1}{10}}) = ln (1000 0^{\frac{9}{10}} \cdot 500 0^{\frac{1}{10}})

= ln (1000 0^{\frac{9}{10}} \cdot 500 0^{\frac{1}{10}})

1000 0^{\frac{9}{10}} \cdot 500 0^{\frac{1}{10}} = 5000 \cdot 2^{\frac{9}{10}}

= ln (5000 \cdot 2^{\frac{9}{10}})

ln (\frac{x}{2}) + 2

lo g_{10} (\frac{1}{5 \cdot 1 0 ^{- 3}})

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

- lo g_{10} (2 x 1 0^{- 5})

2 (lo g_{7} (y) - 2 lo g_{7} (w)) + 3 lo g_{7} (z)

What is 9/10 ln(10000)+1/10 ln(5000) ?
The solution to 9/10 ln(10000)+1/10 ln(5000) is ln(5000*2^{9/10})