Q. Expand the logarithm fully using the properties of logs. Express the final answer in terms of logx.log8x4Answer:
Identify Properties: Identify the properties of logarithms to be used for expansion.We will use the product property and the power property of logarithms to expand log8x4.Product property: logb(mn)=logb(m)+logb(n)Power property: logb(mn)=n⋅logb(m)
Apply Product Property: Apply the product property to separate the constant from the variable.Using the product property, we can write log8x4 as the sum of log8 and logx4.log8x4=log8+logx4
Apply Power Property: Apply the power property to the logarithm of the variable.Using the power property, we can bring the exponent outside the logarithm.logx4=4×logx
Combine Results: Combine the results from Step 2 and Step 3.We combine the logarithm of the constant with the multiple of the logarithm of the variable.log8x4=log8+4⋅logx
Simplify Constant: Simplify the logarithm of the constant.The logarithm of 8 can be simplified since 8 is 23.log8=log(23)=3×log2
Substitute Simplified Form: Substitute the simplified form of log8 into the equation.Replace log8 with 3×log2 in the equation from Step 4.log8x4=3×log2+4×logx