# Writing an expression as a sum or difference of logs??

• Oct 20th 2010, 11:04 PM
yess
Writing an expression as a sum or difference of logs??
Write ln [((x-3)^4(x+5)^1/2)/((2x-7)^9(x^1/3))] as a sum or difference of logarithms with all exponents simplified as far as possible.
• Oct 20th 2010, 11:33 PM
earboth
Quote:

Originally Posted by yess
Write ln [((x-3)^4(x+5)^1/2)/((2x-7)^9(x^1/3))] as a sum or difference of logarithms with all exponents simplified as far as possible.

1. Use the following laws of logs:

$\displaystyle \log(a\cdot b)=\log(a)+\log(b)$

$\displaystyle \log\left(\dfrac ab \right)=\log(a)-\log(b)$

$\displaystyle \log\left(a^n\right)=n\cdot \log(a)$

2. $\displaystyle \ln\left(\dfrac{(x-3)^4 \cdot (x+5)^{\frac12}}{(2x-7)^9 \cdot (x)^{\frac13}} \right) = \ln\left((x-3)^4 \cdot (x+5)^{\frac12}\cdot (2x-7)^{-9} \cdot (x)^{-\frac13} \right)$

$\displaystyle = 4\ln(x-3)+\frac12 \ln(x+5)-9\ln(2x-7)-\frac13 \ln(x)$
• Oct 20th 2010, 11:49 PM
yess
Quote:

Originally Posted by earboth
1. Use the following laws of logs:

$\displaystyle \log(a\cdot b)=\log(a)+\log(b)$

$\displaystyle \log\left(\dfrac ab \right)=\log(a)-\log(b)$

$\displaystyle \log\left(a^n\right)=n\cdot \log(a)$

2. $\displaystyle \ln\left(\dfrac{(x-3)^4 \cdot (x+5)^{\frac12}}{(2x-7)^9 \cdot (x)^{\frac13}} \right) = \ln\left((x-3)^4 \cdot (x+5)^{\frac12}\cdot (2x-7)^{-9} \cdot (x)^{-\frac13} \right)$

$\displaystyle = 4\ln(x-3)+\frac12 \ln(x+5)-9\ln(2x-7)-\frac13 \ln(x)$

great thank you soo much!