What is ln(4x)+3ln(x/2)+ln(2x^2) ?

The solution to ln(4x)+3ln(x/2)+ln(2x^2) is ln(x^6)

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simplify ln(4x)+3ln(x/2)+ln(2x^2)

simplify

ln (4 x) + 3 ln (\frac{x}{2}) + ln (2 x^{2})

ln (x^{6})

Solution steps

ln (4 x) + 3 ln (\frac{x}{2}) + ln (2 x^{2})

Apply log rule:

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

3 ln (\frac{x}{2}) = ln ((\frac{x}{2})^{3})

= ln (4 x) + ln ((\frac{x}{2})^{3}) + ln (2 x^{2})

Apply log rule:

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

ln (4 x) + ln ((\frac{x}{2})^{3}) = ln (4 (\frac{x}{2})^{3} x)

= ln (4 (\frac{x}{2})^{3} x) + ln (2 x^{2})

Simplify

4 x (\frac{x}{2})^{3} : \frac{x ^{4}}{2}

= ln (\frac{x ^{4}}{2}) + ln (2 x^{2})

Apply log rule:

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

ln (\frac{x ^{4}}{2}) + ln (2 x^{2}) = ln (2 \cdot \frac{x ^{4}}{2} x^{2})

= ln (2 \cdot \frac{x ^{4}}{2} x^{2})

Simplify

\frac{x ^{4}}{2} \cdot 2 x^{2} : x^{6}

= ln (x^{6})

ln (\frac{15}{5})

lo g_{10} (9 o)

8 ln (cos^{2} (t)) + 8 ln (1 + tan^{2} (t))

lo g_{4} (9) - 2 lo g_{4} (5)

2 lo g_{6} (5 A) + lo g_{6} (B) + 7 lo g_{6} (C)

What is ln(4x)+3ln(x/2)+ln(2x^2) ?
The solution to ln(4x)+3ln(x/2)+ln(2x^2) is ln(x^6)