Solve for x 3( log base 9 of x)^2=6 log base 9 of x
The given problem is an algebraic equation involving logarithms. Specifically, it is asking to find the value of the variable x that satisfies the equation in which the square of the logarithm of x with base 9, multiplied by 3, is equal to 6 times the logarithm of x with base 9. The form of the equation suggests a quadratic relationship involving the logarithmic function of x. To solve for x, one would typically simplify the equation, potentially turning it into a quadratic equation in terms of the logarithmic expression, and then solving for that expression before finding the value of x itself.
$3 \left(\left(log\right)_{9}\right)^{2} \left(\right. x \left.\right) = 6 \left(log\right)_{9} \left(\right. x \left.\right)$
Answer
Solution:
Step 1:
Distribute the exponent across the logarithm. \(3(\log_{9}{x})^2 = 6\log_{9}{x}\)
Step 2:
Apply the power rule of logarithms to move the coefficient 6 inside the logarithm as an exponent. \(3(\log_{9}{x})^2 = \log_{9}{x^6}\)
Step 3:
Find the value of \(x\) by equating the expressions inside the logarithms, since the bases of the logarithms are the same. \(x = 1\)
Step 4:
There is no further action required as the solution has been found.
Knowledge Notes:
The problem involves solving a logarithmic equation. The key knowledge points include:
Logarithm Properties: Understanding the basic properties of logarithms is crucial. The power rule states that \(a\log_{b}{c} = \log_{b}{c^a}\), which allows you to move a coefficient inside the logarithm as an exponent.
Logarithmic Equations: To solve logarithmic equations, you often need to use properties of logarithms to combine or compare terms.
Graphical Solution: Sometimes, you can solve equations by graphing both sides and finding the intersection points. However, in this problem, the graphical step was not necessary as the algebraic manipulation led to the solution.
Exponent Rules: When dealing with exponents, remember that any number raised to the power of zero is 1, and raising a number to the power of 1 leaves it unchanged. These rules can sometimes simplify the process of solving equations.
Equality of Logarithms: If two logarithms with the same base are equal, then their arguments must also be equal. This principle is used to solve for the variable within the logarithm.
In this problem, the logarithmic equation was simplified using the power rule, and the solution was found by setting the arguments of the logarithms equal to each other since the bases were the same.
