Answer:
The anser is [tex]ln(64y^2)[/tex]
Step-by-step explanation:
- There are two logarithm properties one should take into account to understand that [tex]2ln8+2lny[/tex] and [tex]ln(64y^2)[/tex] are equal expressions.
- The first property refeers to exponentials: [tex]ln(x^n)=n\times{lnx}[/tex].
- The second property is the follow: [tex]ln(x.y)=ln(x)+ln(y)[/tex] (distributive property of the logarithm of a product).
- Applying these two properties to the expression [tex]ln(64y^2)[/tex] results as follows:
- [tex]ln(68y^2)=ln(8y)^2[/tex] because [tex]8\times8=64[/tex]. Then, applying the first property means that [tex]ln(8y)^2=2\times{ln(8y)}[/tex]
- Finally, applying the second property of logarithms to the logarithm of a product implies: [tex]2\times{ln(8y)}=2\times[ln(8)+ln(y)]= 2\times{ln(8)}+2\times{ln(y)}[/tex], which is the expression we needed to arrive.
