What is log_{10}(sqrt(2))+1/2*log_{10}(15)-1/4*log_{10}(9) ?

The solution to log_{10}(sqrt(2))+1/2*log_{10}(15)-1/4*log_{10}(9) is 1/2

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simplify log_{10}(sqrt(2))+1/2log_{10}(15)-1/4log_{10}(9)

simplify

lo g_{10} (2) + \frac{1}{2} \cdot lo g_{10} (15) - \frac{1}{4} \cdot lo g_{10} (9)

\frac{1}{2}

Decimal Notation

0.5

Solution steps

lo g_{10} (2) + \frac{1}{2} lo g_{10} (15) - \frac{1}{4} lo g_{10} (9)

\frac{1}{2} lo g_{10} (15) = lo g_{10} (1 5^{\frac{1}{2}})

- \frac{1}{4} lo g_{10} (9) = - lo g_{10} (9^{\frac{1}{4}})

= lo g_{10} (2) + lo g_{10} (1 5^{\frac{1}{2}}) - lo g_{10} (9^{\frac{1}{4}})

Apply log rule:

lo g_{a} (x) + lo g_{a} (y) = lo g_{a} (x y)

lo g_{10} (2) + lo g_{10} (1 5^{\frac{1}{2}}) = lo g_{10} (2 \cdot 1 5^{\frac{1}{2}})

= lo g_{10} (2 \cdot 1 5^{\frac{1}{2}}) - lo g_{10} (9^{\frac{1}{4}})

Apply log rule:

lo g_{a} (x) - lo g_{a} (y) = lo g_{a} (\frac{x}{y})

lo g_{10} (2 \cdot 1 5^{\frac{1}{2}}) - lo g_{10} (9^{\frac{1}{4}}) = lo g_{10} (\frac{2 \cdot 1 5 ^{\frac{1}{2}}}{9 ^{\frac{1}{4}}})

= lo g_{10} (\frac{2 \cdot 1 5 ^{\frac{1}{2}}}{9 ^{\frac{1}{4}}})

Cancel

\frac{2 \cdot 1 5 ^{\frac{1}{2}}}{9 ^{\frac{1}{4}}} : 10

= lo g_{10} (10)

Apply radical rule:

a = a^{\frac{1}{2}}

10 = 1 0^{\frac{1}{2}}

= lo g_{10} (1 0^{\frac{1}{2}})

Apply log rule:

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

= \frac{1}{2}

lo g_{b} (x^{7} y^{3} z)

- 2 ln ∣ 4 - 1 ∣ + 3 ln ∣ 4 - 2 ∣ + ln 4^{2} + 1 + 2 - ln (\frac{5}{2})

lo g_{2} (5) + lo g_{2} (x^{3})

lo g_{5} (\frac{x ^{11}}{x - 7})

3 e (x y^{3}) - 5 ln (x cos (y))

What is log_{10}(sqrt(2))+1/2*log_{10}(15)-1/4*log_{10}(9) ?
The solution to log_{10}(sqrt(2))+1/2*log_{10}(15)-1/4*log_{10}(9) is 1/2