Log base 8 0.0625

Log base 8 0.0625

The Correct Answer and Explanation is:

To solve log⁡8(0.0625)\log_8(0.0625)log8​(0.0625), we can break the problem into smaller steps by expressing 0.0625 as a power of 8.

Step 1: Express 0.0625 as a fraction

First, express 0.0625 as a fraction: 0.0625=1160.0625 = \frac{1}{16}0.0625=161​

Step 2: Express 16 as a power of 8

We know that: 16=84316 = 8^{\frac{4}{3}}16=834​

because 843=2123=24=168^{\frac{4}{3}} = 2^{\frac{12}{3}} = 2^4 = 16834​=2312​=24=16.

Thus, we can write: 0.0625=18430.0625 = \frac{1}{8^{\frac{4}{3}}}0.0625=834​1​

Step 3: Apply the logarithm rule

Now, let’s apply the logarithm rule: log⁡8(1843)=−log⁡8(843)\log_8\left(\frac{1}{8^{\frac{4}{3}}}\right) = -\log_8\left(8^{\frac{4}{3}}\right)log8​(834​1​)=−log8​(834​)

Using the logarithmic property log⁡b(bx)=x\log_b(b^x) = xlogb​(bx)=x, this becomes: −43-\frac{4}{3}−34​

Final Answer:

Thus, log⁡8(0.0625)=−43\log_8(0.0625) = -\frac{4}{3}log8​(0.0625)=−34​.

Explanation:

Logarithms are the inverse operations of exponents. In this case, log⁡8(0.0625)\log_8(0.0625)log8​(0.0625) is asking: “To what power must we raise 8 to get 0.0625?” By expressing 0.0625 as a fraction and relating it to powers of 8, we find that the answer is −43-\frac{4}{3}−34​, meaning that raising 8 to the power of −43-\frac{4}{3}−34​ gives 0.0625.