Final answer:
To solve the logarithmic equation 2 log3(x) = 4, we use the property that the logarithm of a number raised to an exponent is the product of that exponent and the logarithm of the number, which simplifies to x^2 = 81 and then to the solutions x = 9 and x = -9. However, x must be positive, so the only valid solution is x = 9.
Step-by-step explanation:
To solve the logarithmic equation 2 log3(x) = 4, we apply the properties of logarithms. One key property is that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. Using this property, we can rewrite log3(x2) as 2 log3(x), which matches the left-hand side of the equation, indicating that both sides are equal.
Upon equating 2 log3(x) with 4, we divide both sides by 2 to isolate the logarithm:
log3(x) = 2.
Now, by rewriting the logarithmic equation in exponential form, we have:
x = 32,
which simplifies to:
x2 = 81.
Finally, we take the square root of both sides to find the possible solutions for x:
x = 9, x = -9.
However, because x represents the argument of a logarithm, it must be positive. Therefore, the only valid solution to the original equation is:
x = 9.