what is the solution to the equation below?\nlog20x³ - 2logx = 4\no x = 25\no x = 50\no x = 250\no x = 500

Explanation:

Step1: Apply logarithm property

Use the power - rule of logarithms (n\log_aM=\log_aM^n). So, (2\log x = \log x^2). The equation (\log20x^{3}-2\log x = 4) becomes (\log20x^{3}-\log x^{2}=4). Then, use the quotient - rule (\log_aM-\log_aN=\log_a\frac{M}{N}), and we get (\log\frac{20x^{3}}{x^{2}} = 4). Simplify (\frac{20x^{3}}{x^{2}}) to (20x), so the equation is (\log(20x)=4).

Step2: Convert to exponential form

If the base of the logarithm is 10 (common logarithm), and (\log(20x)=4), 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)