Find the Exact Value 5 log base 7 of sixth root of 49

$5 \log_{7} (\sqrt[6]{49})$

Step 1

Rewrite $49$ as $7^{2}$ .

$5 \log_{7} (\sqrt[6]{7^{2}})$

Step 2

Rewrite $\sqrt[6]{7^{2}}$ as $\sqrt[3]{\sqrt{7^{2}}}$ .

$5 \log_{7} (\sqrt[3]{\sqrt{7^{2}}})$

Step 3

Pull terms out from under the radical, assuming positive real numbers.

$5 \log_{7} (\sqrt[3]{7})$

Step 4

Simplify $5 \log_{7} (\sqrt[3]{7})$ by moving $5$ inside the logarithm.

$\log_{7} ({\sqrt[3]{7}}^{5})$

Step 5

Rewrite ${\sqrt[3]{7}}^{5}$ as ${(7^{5})}^{\frac{1}{3}}$ .

$\log_{7} (\sqrt[3]{7^{5}})$

Step 6

Raise $7$ to the power of $5$ .

$\log_{7} (\sqrt[3]{16807})$

Step 7

Rewrite $16807$ as $7^{3} \cdot 49$ .

Factor $343$ out of $16807$ .

$\log_{7} (\sqrt[3]{343 (49)})$

Rewrite $343$ as $7^{3}$ .

$\log_{7} (\sqrt[3]{7^{3} \cdot 49})$

Step 8

Pull terms out from under the radical.

$\log_{7} (7 \sqrt[3]{49})$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\log_{7} (7 \sqrt[3]{49})$

Decimal Form:

$1.\overline{6}$