Question

which equation is equivalent to \\(\log 5x^3 - \log x^2 = 2\\)?
\\(10^{\log 5x^3} = 10^2\\)
\\(10^{\log \frac{5x^3}{x^2}} = 10^2\\)
\\(10^{\log \frac{5x^3 + x^3}{x^2}} = 10^2\\)
solve: \\(\log 5x^3 - \log x^2 = 2\\)
\\(x = \square\\)

Explanation:

Step1: Apply Logarithm Quotient Rule

The quotient rule of logarithms states that \(\log_a M - \log_a N=\log_a\frac{M}{N}\). For the equation \(\log 5x^3 - \log x^2 = 2\), we can rewrite the left - hand side using the quotient rule. So, \(\log\frac{5x^3}{x^2}=2\). Simplifying the fraction \(\frac{5x^3}{x^2}\), we get \(5x\) (since when dividing variables with exponents, we subtract the exponents: \(x^{3 - 2}=x\)). So the equation becomes \(\log(5x)=2\).

Step2: Convert Logarithmic Equation to Exponential Form

Recall that if \(\log_b y = k\), then \(y = b^k\). Assuming the logarithm is base - 10 (since there is no base specified, it is a common logarithm with base 10), from \(\log(5x)=2\), we can convert it to exponential form. We have \(5x = 10^2\).

Step3: Solve for x

We know that \(10^2 = 100\), so the equation \(5x=100\). To solve for \(x\), we divide both sides of the equation by 5: \(x=\frac{100}{5}=20\).

Answer:

For the equivalent equation part, the correct equivalent equation is \(10^{\log\frac{5x^{3}}{x^{2}}}=10^{2}\) (or simplified \(10^{\log(5x)} = 10^{2}\)). For the solution of \(x\), \(x = 20\)