Question

solve the following logarithmic equation. express irrational solutions in exact form.
$\frac{1}{2} log_{7} x = 3 log_{7} 3$
rewrite the given equation without logarithms. do not solve for x.
(do not simplify. use integers or fractions for any numbers in the equation.)
solve the equation obtained in the previous step for x. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is
(simplify your answer. use a comma to separate answers as needed. use integers or fractions for any numbers in the expression. type an exact answer, using radicals...

Explanation:

Step1: Apply Logarithm Power Rule

The power rule of logarithms states that \( n\log_b a=\log_b a^n \). Apply this to both sides of the equation \(\frac{1}{2}\log_{7}x = 3\log_{7}3\).

For the left - hand side: \(\frac{1}{2}\log_{7}x=\log_{7}x^{\frac{1}{2}}\) (since \(n = \frac{1}{2}\) and \(a=x\), \(b = 7\))

For the right - hand side: \(3\log_{7}3=\log_{7}3^{3}\) (since \(n = 3\) and \(a = 3\), \(b=7\))

So the equation without logarithms (in terms of the arguments of the logarithms, because if \(\log_b m=\log_b n\), then \(m = n\) for \(b>0,b
eq1\)) is \(x^{\frac{1}{2}}=3^{3}\)

Step2: Solve for x

We have the equation \(x^{\frac{1}{2}}=27\) (since \(3^{3}=27\)). To solve for \(x\), we square both sides of the equation.

Squaring both sides: \((x^{\frac{1}{2}})^2=27^{2}\)

Using the exponent rule \((a^m)^n=a^{mn}\), we get \(x = 729\)

Answer:

The solution set is \(\{729\}\)