Is$\log _ { 2 } 5 = \log _ { 16 } 625$

Question

Examlex · Accepted Answer

The Answer is The two quantities are equal. Let

x = \log _ { 16 } 625

. Then, by definition of logarithm,

16 ^ { x } = 625

. Now

16 ^ { x } = \left( 2 ^ { 4 } ight) ^ { x } = \left( 2 ^ { x } ight) ^ { 4 } \quad ext { and } \quad 625 = 5 ^ { 4 } .

So

\left( 2 ^ { x } ight) ^ { 4 } = 5 ^ { 4 } .

Raising both sides to the power

1 / 4

gives that

\left( \left( 2 ^ { x } ight) ^ { 4 } ight) ^ { 1 / 4 } = \left( 5 ^ { 4 } ight) ^ { 1 / 4 } , \quad ext { and thus } 2 ^ { x } = 5 .

Hence, by definition of logarithm,

\log _ { 2 } 5 = x .

Since both quantities equal

x

,

\log _ { 16 } 625 = \log _ { 2 } 5 .

Is $\log _ { 2 } 5 = \log _ { 16 } 625$