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Expand the logarithm fully using the properties of logs. Express the final answer in terms of 
log x.

log 8x^(4)
Answer:

Expand the logarithm fully using the properties of logs. Express the final answer in terms of logx \log x .\newlinelog8x4 \log 8 x^{4} \newlineAnswer:

Full solution

Q. Expand the logarithm fully using the properties of logs. Express the final answer in terms of logx \log x .\newlinelog8x4 \log 8 x^{4} \newlineAnswer:
  1. Identify Properties: Identify the properties of logarithms to be used for expansion.\newlineWe will use the product property and the power property of logarithms to expand log8x4\log 8x^{4}.\newlineProduct property: logb(mn)=logb(m)+logb(n)\log_b(mn) = \log_b(m) + \log_b(n)\newlinePower property: logb(mn)=nlogb(m)\log_b(m^n) = n \cdot \log_b(m)
  2. Apply Product Property: Apply the product property to separate the constant from the variable.\newlineUsing the product property, we can write log8x4\log 8x^{4} as the sum of log8\log 8 and logx4\log x^{4}.\newlinelog8x4=log8+logx4\log 8x^{4} = \log 8 + \log x^{4}
  3. Apply Power Property: Apply the power property to the logarithm of the variable.\newlineUsing the power property, we can bring the exponent outside the logarithm.\newlinelogx4=4×logx\log x^{4} = 4 \times \log x
  4. Combine Results: Combine the results from Step 22 and Step 33.\newlineWe combine the logarithm of the constant with the multiple of the logarithm of the variable.\newlinelog8x4=log8+4logx\log 8x^{4} = \log 8 + 4 \cdot \log x
  5. Simplify Constant: Simplify the logarithm of the constant.\newlineThe logarithm of 88 can be simplified since 88 is 232^3.\newlinelog8=log(23)=3×log2\log 8 = \log (2^3) = 3 \times \log 2
  6. Substitute Simplified Form: Substitute the simplified form of log8\log 8 into the equation.\newlineReplace log8\log 8 with 3×log23 \times \log 2 in the equation from Step 44.\newlinelog8x4=3×log2+4×logx\log 8x^{4} = 3 \times \log 2 + 4 \times \log x

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