What is ln((2x)/3)+1/8 ln(x/4)+3/8 ln((3x)/4) ?

The solution to ln((2x)/3)+1/8 ln(x/4)+3/8 ln((3x)/4) is ln((3^{3/8}x^{11/8}\sqrt[8]{x})/3)

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simplify ln((2x)/3)+1/8 ln(x/4)+3/8 ln((3x)/4)

simplify

ln (\frac{2 x}{3}) + \frac{1}{8} ln (\frac{x}{4}) + \frac{3}{8} ln (\frac{3 x}{4})

Solution steps

ln (\frac{2 x}{3}) + \frac{1}{8} ln (\frac{x}{4}) + \frac{3}{8} ln (\frac{3 x}{4})

Apply log rule:

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

\frac{1}{8} ln (\frac{x}{4}) = ln ((\frac{x}{4})^{\frac{1}{8}})

= ln (\frac{2 x}{3}) + ln ((\frac{x}{4})^{\frac{1}{8}}) + \frac{3}{8} ln (\frac{3 x}{4})

Apply log rule:

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

ln (\frac{2 x}{3}) + ln ((\frac{x}{4})^{\frac{1}{8}}) = ln ((\frac{x}{4})^{\frac{1}{8}} \frac{2 x}{3})

= ln ((\frac{x}{4})^{\frac{1}{8}} \frac{2 x}{3}) + \frac{3}{8} ln (\frac{3 x}{4})

Simplify

Apply log rule:

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

\frac{3}{8} ln (\frac{3 x}{4}) = ln ((\frac{3 x}{4})^{\frac{3}{8}})

Apply log rule:

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

Simplify

ln (a e^{- b x})

- 3 ln (x^{2} + 1) - ln (sin (x))

2 \cdot ln (\frac{2}{3})

lo g_{9} (8 \cdot 7 \cdot 5^{2})

lo g_{10} (\frac{z ^{5} x}{y ^{3}})

What is ln((2x)/3)+1/8 ln(x/4)+3/8 ln((3x)/4) ?
The solution to ln((2x)/3)+1/8 ln(x/4)+3/8 ln((3x)/4) is ln((3^{3/8}x^{11/8}\sqrt[8]{x})/3)