Find the exact value of $\log _ { 5 } \sqrt [ 3 ] { 25 }$ without using a calculator.

A) $\frac { 25 } { 3 }$

B) $\frac { 3 } { 25 }$

C) $\frac { 10 } { 3 }$

D) $\frac { 2 } { 3 }$

E) $- 1$

Question

Find the exact value of $\log _ { 5 } \sqrt [ 3 ] { 25 }$ without using a calculator.

A) $\frac { 25 } { 3 }$

B) $\frac { 3 } { 25 }$

C) $\frac { 10 } { 3 }$

D) $\frac { 2 } { 3 }$

E) $- 1$

Quizplus · Accepted Answer

We can simplify $\sqrt[3]{25}$ as $\sqrt[3]{5^2} = 5^{2/3}$. Then, using the property $\log_a b^n = n \log_a b$, we have:

$$\log_5 \sqrt[3]{25} = \log_5 5^{2/3} = \frac{2}{3} \log_5 5 = \boxed{\frac{2}{3}}$$

Find the Exact Value Of log⁡5253\log _ { 5 } \sqrt [ 3 ] { 25 }log5​325​