expanding log(base a) (2x^2 + 14x + 20) / ( (5)(y^1/3)(z^3))

[F19]

I have to expand:

log(base a) (2x^2 + 14x + 20) / ( (5)(y^1/3)(z^3) )

*Do I have to factor (2x^2 + 14x + 20) before I do anything else?

Thanks alot!

(Please show some work and the final answer)

 

[arthur ohlsten]

Re: Help with expanding logarithms?

I believe you should to obtain a product and not sums
log is log to base a

log[(2x^2+14x+20) / (5y^1/3z^3) ]

first step is easy division becomes subtraction of logs
log[2x^2+14x+20] - log [5 y^1/3 z^3]

let us expand the second part first
log[5y^1/3z^3]=log 5 +1/3 logy +3 log z part answer

econd part. we will first factor out 2
log[2x^2+14x+20]= log 2 [x^2+7x+10]
log[2x^2+14x+20]=log 2 + log [x^2+7x+10]

but we want products or divisions, not sums , we shall factor as you suggested
log[2x^2+14x+20]=log 2 + log[[x+5][x+2]]
log [2x^2+14x+20]=log2 + log[x+5]+log [x+2]

you can now finish the problem

Arthur

 

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... Please show some work and the final answer

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... you can now finish the problem

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[soroban]

Hello, F19!

\(\displaystyle \text{Expand: }\;\log_a\bigg[\frac{2x^2 + 14x + 20}{5y^{\frac{1}{3}}z^3}\bigg]\)

Do I have to factor \(\displaystyle 2x^2 + 14x + 20\) before I do anything else? . . . . I would

\(\displaystyle \text{We have: }\;\log_a\left[\frac{2(x+2)(x+5)}{5y^{\frac{1}{3}}z^3}\right] \;=\;\log_a\bigg[2(x+2)(x+5)\bigg] - \log_a\bigg[5y^{\frac{1}{3}}z^3\bigg]\)

. . \(\displaystyle = \;\bigg[\log_a(2) + \log_a(x+2) + \log_a(x+5)\bigg] - \bigg[\log_a(5) + \log_a(y^{\frac{1}{3}}) + \log_a(z^3)\bigg]\)

. . \(\displaystyle = \;\log_a(2) + \log_a(x+2) + \log_a(x+5) - \log_a(5) - \log_a(y^{\frac{1}{3}}) - \log_a(z^3)\)


. . \(\displaystyle = \;\log_a(2) + \log_a(x+2) + \log_a(x+5) - \log_a(5) - \tfrac{1}{3}\log_a(y) - 3\log_a(z)\)