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

The solution to ln(4)+1/3 ln(8) is 3ln(2)

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

simplify

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

3 ln (2)

Decimal Notation

2.07944 \dots

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

= ln (4 \cdot 8^{\frac{1}{3}})

4 \cdot 8^{\frac{1}{3}} = 8

= ln (8)

Rewrite

8

in power-base form:

8 = 2^{3}

= ln (2^{3})

Apply log rule:

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

ln (2^{3}) = 3 ln (2)

= 3 ln (2)

\frac{1}{2} lo g_{10} (c) + 3 lo g_{10} (d)

- \frac{5}{7} lo g_{2} (\frac{5}{7}) - \frac{2}{7} lo g_{2} (\frac{2}{7})

lo g_{10} (\frac{x ^{5}}{y ^{4}})

ln (\frac{x + 9}{x ( x - 4 )})

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