Solve for x 2 log base 3 of x=3 log base 3 of 5

Problem

2*(log_3)(x)=3*(log_3)(5)

Solution

1. Apply the power property of logarithms, which states n*(log_b)(a)=(log_b)(an) to both sides of the equation.

(log_3)(x2)=(log_3)(5)

2. Simplify the exponent on the right side of the equation by calculating the value of 5

(log_3)(x2)=(log_3)(125)

3. Use the one-to-one property of logarithms, which states if (log_b)(M)=(log_b)(N) then M=N

x2=125

4. Solve for x by taking the square root of both sides.

x=√(,125)

5. Simplify the radical by factoring out the largest perfect square.

x=5√(,5)

6. Check for extraneous solutions by ensuring the argument of the original logarithm, x is greater than zero. Since 5√(,5)>0 the solution is valid. Note that the negative root x=−5√(,5) is excluded because the domain of (log_3)(x) is x>0

x=5√(,5)

Final Answer

x=5√(,5)

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