What is log_{4}(x^5)-log_{4}(x^3) ?

The solution to log_{4}(x^5)-log_{4}(x^3) is log_{2}(x)

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simplify log_{4}(x^5)-log_{4}(x^3)

simplify

lo g_{4} (x^{5}) - lo g_{4} (x^{3})

lo g_{2} (x)

Solution steps

lo g_{4} (x^{5}) - lo g_{4} (x^{3})

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x), assuming x > 0

lo g_{4} (x^{5}) = 5 lo g_{4} (x)

= 5 lo g_{4} (x) - lo g_{4} (x^{3})

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x), assuming x > 0

lo g_{4} (x^{3}) = 3 lo g_{4} (x)

= 5 lo g_{4} (x) - 3 lo g_{4} (x)

Add similar elements:

5 lo g_{4} (x) - 3 lo g_{4} (x) = 2 lo g_{4} (x)

= 2 lo g_{4} (x)

Factor the number:

4 = 2^{2}

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

Apply log rule:

lo g_{a^{b}} (x) = \frac{1}{b} lo g_{a} (x), a > 0

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

= 2 \cdot \frac{1}{2} lo g_{2} (x)

Simplify

2 \cdot \frac{1}{2} lo g_{2} (x) : lo g_{2} (x)

= lo g_{2} (x)

3 ln (x) - \frac{2}{3} ln (x + 1) + ln (y) - 3 ln (z)

3 lo g_{7} (2 x) - 2 lo g_{7} (3 y)

\frac{1}{2} lo g_{2} (6) - 2 lo g_{2} (5)

\frac{1}{2} lo g_{2} (6) - 2 lo g_{2} (7)

3 ln (x^{3} y) + 2 ln (y z^{2})

What is log_{4}(x^5)-log_{4}(x^3) ?
The solution to log_{4}(x^5)-log_{4}(x^3) is log_{2}(x)