Question

Expand the following logs:

$log_{7} \sqrt{a^{3}b^{9} }$

$log_{6} (\frac{x^{5} }{y^{9} } )$

$log_{8} (x^{3} y^{7} )$

Answer 1

QUESTION 1

The given logarithm is

$\log_7\sqrt{a^3b^9}$

We rewrite to obtain;

$\log_7(a^3b^9)^{\frac{1}{2}}$

Use the power rule of logarithm ; $\log_a(m^n)=n\log_a(m)$

$\frac{1}{2}\log_7(a^3b^9)$

Use the product rule; $\log_a(mn)=\log_a(m)+\log_a(n)$

$\frac{1}{2}[\log_7(a^3)+\log_7(b^9)]$

Use the power rule of logarithms again;

$\frac{1}{2}[3\log_7(a)+9\log_7(b)]$

Or

$\frac{3}{2}\log_7(a)+\frac{9}{2}\log_7(b)]$

QUESTION 2

Given;

$\log_6(\frac{x^5}{y^9})$

Apply the quotient rule of logarithm; $\log_a(m)-\log_a(n)=\log_a(\frac{m}{n} )$

$\log_6(\frac{x^5}{y^9})=\log_6(x^5)-\log_6(y^9)$

Apply the power rule to get;

$\log_6(\frac{x^5}{y^9})=5\log_6(x)-9\log_6(y)$

QUESTION 3

Given;

$\log_8(x^3y^7)$

Use the product rule to get;

$=\log_8(x^3)+\log_8(y^7)$

Use the power rule now;

$=3\log_8(x)+7\log_8(y)$