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Given b > 0, simplify each of the following: a) (i)log_(b)b^(4)

Given b>0 b>0 , simplify the following:\newline log⁡bb4 \log _{b} b^{4}

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Full solution

Q. Given b>0 b>0 , simplify the following:\newline log⁡bb4 \log _{b} b^{4}
  1. Identify Property: Identify the property used to simplify log⁡b(b4)\log_b(b^4). \newlineUsing the power property of logarithms, which states log⁡b(mn)=n⋅log⁡b(m)\log_b(m^n) = n \cdot \log_b(m), we can simplify the expression.
  2. Apply Power Property: Apply the power property to log⁡b(b4)\log_b(b^4).\newlinelog⁡b(b4)=4⋅log⁡b(b)\log_b(b^4) = 4 \cdot \log_b(b)
  3. Simplify Logarithm: Simplify log⁡b(b)\log_b(b). \newlineSince log⁡b(b)\log_b(b) is the logarithm of the base to itself, it equals 11. \newlinelog⁡b(b)=1\log_b(b) = 1
  4. Multiply Values: Multiply the values.\newline4×log⁡b(b)=4×1=44 \times \log_b(b) = 4 \times 1 = 4

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