what is the solution to the equation below?\nlog20x³ - 2logx = 4\nx = 25\nx = 50\nx = 250\nx = 500

Explanation:

Step1: Apply logarithm rules

Using the power - rule of logarithms (n\log_aM=\log_aM^n) and the quotient - rule (\log_aM-\log_aN = \log_a\frac{M}{N}), we rewrite the left - hand side of the equation. (\log(20x^{3})-2\log x=\log(20x^{3})-\log(x^{2})=\log\frac{20x^{3}}{x^{2}}=\log(20x)) So the equation becomes (\log(20x) = 4).

Step2: Convert from logarithmic to exponential form

If (\log_{10}(20x)=4) (assuming base 10 for the common logarithm), then by the definition of logarithms (y = \log_a x) is equivalent to (a^{y}=x). Here (a = 10), (y = 4), and (x = 20x). So (10^{4}=20x).

Step3: Solve for (x)

We know that (10^{4}=10000), so the equation (10000 = 20x). Divide both sides by 20: (x=\frac{10000}{20}=500).

Answer:

(x = 500)