Question

solve the equation. 5 log x - log x⁴ = 1 select the correct choice below and fill in any answer boxes present in your choice. a. x = (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. use a comma to separate answers b. there is no solution.

Explanation:

Step1: Apply log power rule

Using the power rule of logarithms, \(n\log_a b=\log_a b^n\), we can rewrite \(5\log x\) as \(\log x^5\). So the equation \(5\log x - \log x^4 = 1\) becomes \(\log x^5-\log x^4 = 1\).

Step2: Apply log quotient rule

Using the quotient rule of logarithms, \(\log_a M-\log_a N=\log_a\frac{M}{N}\), we have \(\log\frac{x^5}{x^4}=1\). Simplifying the argument of the logarithm, \(\frac{x^5}{x^4}=x\) (for \(x
eq0\)), so the equation is now \(\log x = 1\).

Step3: Convert to exponential form

Assuming the logarithm is base 10 (since no base is specified, we use base 10), we convert \(\log x = 1\) to exponential form. Recall that \(\log_{10} x = y\) is equivalent to \(10^y=x\). So here, \(y = 1\), so \(x = 10^1=10\). We need to check if this solution is valid. The domain of the original logarithmic functions: for \(\log x\) and \(\log x^4\), \(x>0\). When \(x = 10\), both \(\log 10\) and \(\log 10^4\) are defined (since \(10>0\) and \(10^4>0\)). So \(x = 10\) is a valid solution.

Answer:

A. \(x = 10\)