What is 1/2 log_{b}(0.35)-1/3 log_{b}(0.85) ?

The solution to 1/2 log_{b}(0.35)-1/3 log_{b}(0.85) is log_{b}(0.62454…)

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simplify 1/2 log_{b}(0.35)-1/3 log_{b}(0.85)

simplify

\frac{1}{2} lo g_{b} (0.35) - \frac{1}{3} lo g_{b} (0.85)

lo g_{b} (0.62454 \dots)

Solution steps

\frac{1}{2} lo g_{b} (0.35) - \frac{1}{3} lo g_{b} (0.85)

Apply log rule:

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

\frac{1}{2} lo g_{b} (0.35) = lo g_{b} (0.3 5^{\frac{1}{2}})

= lo g_{b} (0.3 5^{\frac{1}{2}}) - \frac{1}{3} lo g_{b} (0.85)

Apply log rule:

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

\frac{1}{3} lo g_{b} (0.85) = lo g_{b} (0.8 5^{\frac{1}{3}})

= lo g_{b} (0.3 5^{\frac{1}{2}}) - lo g_{b} (0.8 5^{\frac{1}{3}})

Apply log rule:

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

lo g_{b} (0.3 5^{\frac{1}{2}}) - lo g_{b} (0.8 5^{\frac{1}{3}}) = lo g_{b} (\frac{0.3 5 ^{\frac{1}{2}}}{0.8 5 ^{\frac{1}{3}}})

= lo g_{b} (\frac{0.3 5 ^{\frac{1}{2}}}{0.8 5 ^{\frac{1}{3}}})

\frac{0.3 5 ^{\frac{1}{2}}}{0.8 5 ^{\frac{1}{3}}} = 0.62454 \dots

= lo g_{b} (0.62454 \dots)

\frac{9}{2} ln (60 - x) + ln (30 - x)

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What is 1/2 log_{b}(0.35)-1/3 log_{b}(0.85) ?
The solution to 1/2 log_{b}(0.35)-1/3 log_{b}(0.85) is log_{b}(0.62454…)