What is 3/8*ln(8/3)+10/16*ln(16/10) ?

The solution to 3/8*ln(8/3)+10/16*ln(16/10) is 1/8 ln(16777216/84375)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

simplify 3/8ln(8/3)+10/16ln(16/10)

simplify

\frac{3}{8} \cdot ln (\frac{8}{3}) + \frac{10}{16} \cdot ln (\frac{16}{10})

\frac{1}{8} ln (\frac{16777216}{84375})

Decimal Notation

0.66156 \dots

Solution steps

\frac{3}{8} ln (\frac{8}{3}) + \frac{10}{16} ln (\frac{16}{10})

Apply log rule:

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

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

= ln ((\frac{8}{3})^{\frac{3}{8}}) + \frac{10}{16} ln (\frac{16}{10})

Apply log rule:

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

\frac{10}{16} ln (\frac{16}{10}) = ln ((\frac{16}{10})^{\frac{10}{16}})

= ln ((\frac{8}{3})^{\frac{3}{8}}) + ln ((\frac{16}{10})^{\frac{10}{16}})

Apply log rule:

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

ln ((\frac{8}{3})^{\frac{3}{8}}) + ln ((\frac{16}{10})^{\frac{10}{16}}) = ln ((\frac{8}{3})^{\frac{3}{8}} (\frac{16}{10})^{\frac{10}{16}})

= ln ((\frac{8}{3})^{\frac{3}{8}} (\frac{16}{10})^{\frac{10}{16}})

(\frac{16}{10})^{\frac{10}{16}} = (\frac{8}{5})^{\frac{5}{8}}

= ln ((\frac{8}{3})^{\frac{3}{8}} (\frac{8}{5})^{\frac{5}{8}})

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

Rewrite as

= ln ((\frac{8}{3})^{3} (\frac{8}{5})^{5})^{\frac{1}{8}}

Apply log rule

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

assuming

x \geq 0

= \frac{1}{8} ln ((\frac{8}{5})^{5} (\frac{8}{3})^{3})

(\frac{8}{3})^{3} = \frac{8 ^{3}}{3 ^{3}}

= \frac{1}{8} ln ((\frac{8}{5})^{5} \frac{8 ^{3}}{3 ^{3}})

(\frac{8}{5})^{5} = \frac{8 ^{5}}{5 ^{5}}

= \frac{1}{8} ln (\frac{8 ^{3}}{3 ^{3}} \cdot \frac{8 ^{5}}{5 ^{5}})

Multiply

\frac{8 ^{3}}{3 ^{3}} \cdot \frac{8 ^{5}}{5 ^{5}} : \frac{16777216}{84375}

= \frac{1}{8} ln (\frac{16777216}{84375})

ln (\frac{3 ^{n}}{2 ^{n} b ^{n}})

\frac{6}{5} ln ∣ z - 1 ∣ - \frac{1}{5} ln ∣ z + 4 ∣ + ln ∣ x ∣

ln (2) + \frac{1}{2} ln (5)

2000 ln (\frac{140000}{140000 - 4200}) - (9.8 \cdot 2)

lo g_{b} (\frac{a ^{\frac{- 3}{2}}}{b ^{4}})

What is 3/8*ln(8/3)+10/16*ln(16/10) ?
The solution to 3/8*ln(8/3)+10/16*ln(16/10) is 1/8 ln(16777216/84375)