What is 3/2 log_{10}(3)+3/2 log_{10}(2) ?

The solution to 3/2 log_{10}(3)+3/2 log_{10}(2) is 3/2 log_{10}(6)

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simplify 3/2 log_{10}(3)+3/2 log_{10}(2)

simplify

\frac{3}{2} lo g_{10} (3) + \frac{3}{2} lo g_{10} (2)

\frac{3}{2} lo g_{10} (6)

Decimal Notation

1.16722 \dots

Solution steps

\frac{3}{2} lo g_{10} (3) + \frac{3}{2} lo g_{10} (2)

Apply log rule:

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

\frac{3}{2} lo g_{10} (3) = lo g_{10} (3^{\frac{3}{2}})

= lo g_{10} (3^{\frac{3}{2}}) + \frac{3}{2} lo g_{10} (2)

Apply log rule:

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

\frac{3}{2} lo g_{10} (2) = lo g_{10} (2^{\frac{3}{2}})

= lo g_{10} (3^{\frac{3}{2}}) + lo g_{10} (2^{\frac{3}{2}})

Apply log rule:

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

lo g_{10} (3^{\frac{3}{2}}) + lo g_{10} (2^{\frac{3}{2}}) = lo g_{10} (3^{\frac{3}{2}} \cdot 2^{\frac{3}{2}})

= lo g_{10} (3^{\frac{3}{2}} \cdot 2^{\frac{3}{2}})

3^{\frac{3}{2}} \cdot 2^{\frac{3}{2}} = 66

= lo g_{10} (66)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

66 = 6^{1 + \frac{1}{2}} = 6^{\frac{3}{2}}

= lo g_{10} (6^{\frac{3}{2}})

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

assuming

x \geq 0

= \frac{3}{2} lo g_{10} (6)

lo g_{10} (14 x)

lo g_{10} (\frac{z ^{3}}{100})

ln (\frac{x}{yz})

ln (\frac{e ^{4}}{7})

lo g_{10} (\frac{5 b}{10})

What is 3/2 log_{10}(3)+3/2 log_{10}(2) ?
The solution to 3/2 log_{10}(3)+3/2 log_{10}(2) is 3/2 log_{10}(6)