What is log_{2}(a)+log_{2}(25)-log_{4}((5a)^2) ?

The solution to log_{2}(a)+log_{2}(25)-log_{4}((5a)^2) is log_{2}(5)

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simplify log_{2}(a)+log_{2}(25)-log_{4}((5a)^2)

simplify

lo g_{2} (a) + lo g_{2} (25) - lo g_{4} ((5 a)^{2})

lo g_{2} (5)

Decimal Notation

2.32192 \dots

Solution steps

lo g_{2} (a) + lo g_{2} (25) - lo g_{4} ((5 a)^{2})

Apply log rule:

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

lo g_{2} (a) + lo g_{2} (25) = lo g_{2} (a \cdot 25)

= lo g_{2} (a \cdot 25) - lo g_{4} ((5 a)^{2})

Simplify

lo g_{4} ((5 a)^{2}) : lo g_{2} (5 a)

= lo g_{2} (a \cdot 25) - lo g_{2} (5 a)

Apply log rule:

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

lo g_{2} (a \cdot 25) - lo g_{2} (5 a) = lo g_{2} (\frac{a \cdot 25}{5 a})

= lo g_{2} (\frac{a \cdot 25}{5 a})

Cancel

\frac{a \cdot 25}{5 a} : 5

= lo g_{2} (5)

lo g_{a} (mn)

ln (2 x + 1) + ln (x + 2)

4 lo g_{3} (x) - 20 lo g_{3} (y)

2 lo g_{10} (a) + 5 lo g_{10} (b)

ln (7) - ln (2)

What is log_{2}(a)+log_{2}(25)-log_{4}((5a)^2) ?
The solution to log_{2}(a)+log_{2}(25)-log_{4}((5a)^2) is log_{2}(5)