Answer:
D. [tex]\log_4(3\sqrt[3]{35})[/tex]
Step-by-step explanation:
The given logarithmic expression is:
[tex]\log_43+\frac{\log_45}{3} +\frac{\log_47}{3}[/tex]
Don't let the fractions scare you at all.
We can rewrite the expression in another form that makes the fractions a bit friendly.
Recall that: [tex]\boxed{\frac{x}{3}=\frac{1}{3}x}[/tex]
We apply this knowledge to get:
[tex]\log_43+\frac{1}{3}\log_45 +\frac{1}{3}\log_47[/tex]
We can now use the following property:
[tex]n \log_am=\log_am^n[/tex]
We apply this property to get:
[tex]\log_43+\log_45^{\frac{1}{3}} +\frac{1}{3}\log_47^{\frac{1}{3}}[/tex]
Recall again that:
[tex]\log_am+\log_an+\log_ap=\log_amnp[/tex]
[tex]\log_43\times 5^{\frac{1}{3}} \times7^{\frac{1}{3}}[/tex]
[tex]\log_43\times (5\times7)^{\frac{1}{3}}[/tex]
[tex]\log_43\times (35)^{\frac{1}{3}}[/tex]
[tex]\log_43\sqrt[3]{35}[/tex]
The correct choice is D.
