Log3(x)=log9(6x)
Not very patient, are ya?
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To solve the equation log3(x) = log9(6x), we can use the logarithmic identity log_a(b) = log_c(b) / log_c(a), where a, b, and c are positive real numbers.
Applying this identity to our equation, we have:
log3(x) = log9(6x)
log(x) / log(3) = log(6x) / log(9)
Now, we need to simplify the equation. Firstly, we can rewrite the logarithms on the right side of the equation using the base conversion formula, log_a(b) = log_c(b) / log_c(a):
log(x) / log(3) = log(6x) / (log(3^2))
We can further simplify the right side of the equation by using the logarithmic properties. Since log(a * b) = log(a) + log(b), we can rewrite log(3^2) as 2 * log(3):
log(x) / log(3) = log(6x) / (2 * log(3))
Next, we can eliminate the denominators by cross-multiplying:
log(x) * (2 * log(3)) = log(6x) * log(3)
Simplifying further gives:
2 * log(x) * log(3) = log(6x) * log(3)
Now, we can cancel out the common factor log(3) from both sides of the equation:
2 * log(x) = log(6x)
Next, we can use the logarithmic property log(a^b) = b * log(a) to rewrite the equation as:
log(x^2) = log(6x)
Since the logarithmic function is one-to-one, if two logarithmic expressions with the same base are equal, then their arguments are equal. Therefore, we have:
x^2 = 6x
This is now a quadratic equation, so let's bring all the terms to one side of the equation:
x^2 - 6x = 0
Factoring out an x from the left side of the equation gives us:
x(x - 6) = 0
Setting each factor to zero, we have two possible solutions:
x = 0 or x - 6 = 0
Solving for x in the second equation, we find:
x = 6
Therefore, the solutions to the equation log3(x) = log9(6x) are x = 0 and x = 6.