tonan6e

2022-09-01

Need help with expressing this logarithm

Express $lo{g}_{3}({a}^{2}+\sqrt{b})$ in terms of m and k where $m=lo{g}_{3}a$

$k=lo{g}_{3}b$

Given this information I made $a={3}^{m}$

$b={3}^{k}$

Therefore = $lo{g}_{3}(({3}^{m}{)}^{2}+({3}^{k}){)}^{\frac{1}{2}}$

= $lo{g}_{3}({3}^{2m}+{3}^{\frac{k}{2}})$

I don't know if I'm done or there is still more things I can simplify. Can anyone help please, thanks

Express $lo{g}_{3}({a}^{2}+\sqrt{b})$ in terms of m and k where $m=lo{g}_{3}a$

$k=lo{g}_{3}b$

Given this information I made $a={3}^{m}$

$b={3}^{k}$

Therefore = $lo{g}_{3}(({3}^{m}{)}^{2}+({3}^{k}){)}^{\frac{1}{2}}$

= $lo{g}_{3}({3}^{2m}+{3}^{\frac{k}{2}})$

I don't know if I'm done or there is still more things I can simplify. Can anyone help please, thanks

Phoenix Owen

Beginner2022-09-02Added 6 answers

Since there is no expansion for $\mathrm{log}(a+b)$, hence, you could stop there. If you need to get an approximation(sometimes in computer science), you could use $\mathrm{log}(a+b)=\mathrm{log}a+\mathrm{log}(1+b/a)$, then you have following:

$$2m+{\mathrm{log}}_{3}(1+{3}^{k/2-2m}).$$

$$2m+{\mathrm{log}}_{3}(1+{3}^{k/2-2m}).$$

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