What is log_{3}(3/2)+log_{3}(4.5) ?

The solution to log_{3}(3/2)+log_{3}(4.5) is 3-2log_{3}(2)

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simplify log_{3}(3/2)+log_{3}(4.5)

simplify

lo g_{3} (\frac{3}{2}) + lo g_{3} (4.5)

3 - 2 lo g_{3} (2)

Decimal Notation

1.73814 \dots

Solution steps

lo g_{3} (\frac{3}{2}) + lo g_{3} (4.5)

Apply log rule:

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

lo g_{3} (\frac{3}{2}) + lo g_{3} (4.5) = lo g_{3} (\frac{3}{2} \cdot 4.5)

= lo g_{3} (\frac{3}{2} \cdot 4.5)

\frac{3}{2} \cdot 4.5 = \frac{27}{4}

= lo g_{3} (\frac{27}{4})

Apply log rule:

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

lo g_{3} (\frac{27}{4}) = lo g_{3} (27) - lo g_{3} (4)

= lo g_{3} (27) - lo g_{3} (4)

Simplify

lo g_{3} (27) : 3

Rewrite

4

in power-base form:

4 = 2^{2}

= 3 - lo g_{3} (2^{2})

Apply log rule:

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

lo g_{3} (2^{2}) = 2 lo g_{3} (2)

= 3 - 2 lo g_{3} (2)

\frac{1}{2} lo g_{10} (x) - lo g_{10} (3 y)

ln (\frac{9310}{3045})

2 lo g_{10} (x) - lo g_{10} (7)

\frac{1}{6} [5 ln (x + 7) - ln (x) - ln (x^{2} - 4)]

ln (\frac{( x - 3 ) ^{2} ( x + 2 ) ^{4}}{( 5 x - 1 ) ^{6}})

What is log_{3}(3/2)+log_{3}(4.5) ?
The solution to log_{3}(3/2)+log_{3}(4.5) is 3-2log_{3}(2)