log base 8 (0.25) compute the log

log base 8 (0.25) compute the log

The Correct Answer and Explanation is:

To compute:log⁡8(0.25)\log_8(0.25)log8​(0.25)

Step 1: Change of base formula
Use the logarithmic identity:log⁡b(a)=log⁡c(a)log⁡c(b)\log_b(a) = \frac{\log_c(a)}{\log_c(b)}logb​(a)=logc​(b)logc​(a)​

Let base c=10c = 10c=10:log⁡8(0.25)=log⁡10(0.25)log⁡10(8)\log_8(0.25) = \frac{\log_{10}(0.25)}{\log_{10}(8)}log8​(0.25)=log10​(8)log10​(0.25)​

Step 2: Calculate the values
Use a calculator or log tables:log⁡10(0.25)≈−0.6021log⁡10(8)=log⁡10(23)=3log⁡10(2)≈3×0.3010=0.9030\log_{10}(0.25) \approx -0.6021 \\ \log_{10}(8) = \log_{10}(2^3) = 3\log_{10}(2) \approx 3 \times 0.3010 = 0.9030log10​(0.25)≈−0.6021log10​(8)=log10​(23)=3log10​(2)≈3×0.3010=0.9030

Now divide:log⁡8(0.25)≈−0.60210.9030≈−0.6667\log_8(0.25) \approx \frac{-0.6021}{0.9030} \approx -0.6667log8​(0.25)≈0.9030−0.6021​≈−0.6667

Final Answer:log⁡8(0.25)≈−23\log_8(0.25) \approx -\frac{2}{3}log8​(0.25)≈−32​

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Explanation

The logarithmic expression log⁡8(0.25)\log_8(0.25)log8​(0.25) asks for the exponent to which 8 must be raised to yield 0.25. Since 8 is greater than 1 and 0.25 is less than 1, the result must be negative.

To simplify computation, the change-of-base formula proves useful. This identity allows conversion of a logarithm with any base into a ratio of logarithms with a common base such as 10 or eee. Applying this identity:log⁡8(0.25)=log⁡10(0.25)log⁡10(8)\log_8(0.25) = \frac{\log_{10}(0.25)}{\log_{10}(8)}log8​(0.25)=log10​(8)log10​(0.25)​

Using decimal approximations:

• log⁡10(0.25)≈−0.6021\log_{10}(0.25) \approx -0.6021log10​(0.25)≈−0.6021, since 0.25 equals 14\frac{1}{4}41​, and logarithms of values between 0 and 1 are negative.
• log⁡10(8)=log⁡10(23)=3log⁡10(2)≈3×0.3010=0.9030\log_{10}(8) = \log_{10}(2^3) = 3\log_{10}(2) \approx 3 \times 0.3010 = 0.9030log10​(8)=log10​(23)=3log10​(2)≈3×0.3010=0.9030

Thus,log⁡8(0.25)≈−0.60210.9030≈−0.6667\log_8(0.25) \approx \frac{-0.6021}{0.9030} \approx -0.6667log8​(0.25)≈0.9030−0.6021​≈−0.6667

The decimal −0.6667-0.6667−0.6667 equals the fraction −23-\frac{2}{3}−32​. This means that:8−2/3=0.258^{-2/3} = 0.258−2/3=0.25

To verify, rewrite 8 as 232^323:(23)−2/3=2−2=14=0.25(2^3)^{-2/3} = 2^{-2} = \frac{1}{4} = 0.25(23)−2/3=2−2=41​=0.25

This confirms the result is exact. The logarithm evaluates to −23-\frac{2}{3}−32​.