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Apply the logarithm property b x - b y = _b ((x)/(y)) to the first two terms.

Mathematics

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Answer

83\frac{8}{3}

Step-by-step solution

5 steps

  1. Apply the logarithm property log⁡bx−log⁡by=log⁡b(xy)\log_b x - \log_b y = \log_b \left(\frac{x}{y}\right) to the first two terms.

    log⁡82−log⁡80.25+log⁡832=log⁡8(20.25)+log⁡832\log_8 2 - \log_8 0.25 + \log_8 32 = \log_8 \left(\frac{2}{0.25}\right) + \log_8 32

  2. Simplify the argument of the first logarithm.

    log⁡8(20.25)=log⁡8(214)=log⁡8(2×4)=log⁡88\log_8 \left(\frac{2}{0.25}\right) = \log_8 \left(\frac{2}{\frac{1}{4}}\right) = \log_8 (2 \times 4) = \log_8 8

    The expression becomes:

    log⁡88+log⁡832\log_8 8 + \log_8 32

  3. Apply the logarithm property log⁡bx+log⁡by=log⁡b(x⋅y)\log_b x + \log_b y = \log_b (x \cdot y) to the remaining terms.

    log⁡88+log⁡832=log⁡8(8×32)\log_8 8 + \log_8 32 = \log_8 (8 \times 32)

  4. Simplify the argument of the logarithm.

    log⁡8(8×32)=log⁡8256\log_8 (8 \times 32) = \log_8 256

  5. Evaluate the logarithm.

    Let x=log⁡8256x = \log_8 256. This means 8x=2568^x = 256.
    We can express both 8 and 256 as powers of 2:

    8=238 = 2^3 256=28256 = 2^8

    Substitute these into the equation:

    (23)x=28(2^3)^x = 2^8 23x=282^{3x} = 2^8

    Equating the exponents:

    3x=83x = 8 x=83x = \frac{8}{3}

    The final answer is 83\frac{8}{3}.

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