logarithmic prob

[kpx001]

simplifying
log7sqroot[x^4y^7]



log7x^2(y)
2log7x+log7y


thats what i got but its wrong. the log base needs to b 2. can anyone show step / process?

 

[soroban]

Hello, kpx001!

Simplify: \(\displaystyle \:\log_7\sqrt{x^4y^7}\)

We have: \(\displaystyle \,\log_7\left(x^4y^7\right)^{\frac{1}{2}}\;=\;\frac{1}{2}\cdot\log\left(x^4y^7\right) \;= \;\frac{1}{2}\left[\log\left(x^4\right)\,+\,\log_7\left(y^7\right)\right] \;=\;\frac{1}{2}\left[4\cdot\log_7(x)\,+\,7\cdot\log_7(y)\right]\)

Answer: \(\displaystyle \,2\cdot\log_7(x)\,+\,\frac{7}{2}\cdot\log_7(y)\)

The log base needs to be 2.

Now you tell us! . . . Use the Base-Change Formula.

Answer: \(\displaystyle \L\,2\cdot\frac{\log_2(x)}{\log_2(7)}\,+\,\frac{7}{2}\cdot\frac{\log_2(y)}{\log_2(7)}\)