Value of log 16 base 4 | Find log_4 16

The value of log 16 base 4 is equal to 2, that is, log₄16 = 2. The logarithm of 16 to the base is expressed as log₄16, and its formula is given by

$\boxed{\log_4 16 =2}$.

From the definition, we know that x = log_ab if and only if a^x=b. Therefore, if the x is the value of log₄16, that is, if

x=log₄16

then we must have 4^x=16. This is valid when x=2. So log 16 base 4 is equal to 4. Let us now explain the above in more details.

Find log 16 to the base 4

Question: What is the value of log16 to the base 4?

Answer: The value of logarithm of 16 to the base 4 is 2.

Explanation:

We know that

16 = 4×4

⇒ 16 = 4²

Now, taking logarithms with base 4 on both sides, we get that

log416 = log442 ⇒ log416 = 2 log44 as we know logabn = n logab. ⇒ log416 = 2 × 1, because logaa =1. ⇒ log416 = 2.

So the value of log 16 to the base 4 is equal to 2.

Question: Using log₄16 = 2, find the value of log₄256.

Answer:

Observe that 256 is a perfect square, which is a square root of 16, i.e, 256 = 16². Now, using the property log_ab^k = k log_ab, we have that

log4256 = log4162

⇒ log4256 = 2 × log416

⇒ log4256 = 2 × 2 as we know from above that log416 = 2.

⇒ log4256 = 4.

So the value of log256 with base 4 is equal to 4.

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