log (base 3) x^2 = 2log(base 3)4-4log(base 3)5
And I need to solve for x.
assume all logs are base 3. We have
log x^2 = 2log4 - 4log5
log x^2 = log16 - log625
x^2 = 16/625
x = ±4/25
Yes, the -4/25 is a solution, since only x^2 appears, so it's always positive.
To solve the given equation log(base 3) x^2 = 2log(base 3)4 - 4log(base 3)5 for x, we can simplify both sides of the equation separately by using rules of logarithms.
First, let's simplify the right side of the equation:
2log(base 3)4 - 4log(base 3)5
Using the power rule of logarithms, we can rewrite log(base 3)4^2 as 2log(base 3)4:
2log(base 3)4 - 4log(base 3)5 = log(base 3)4^2 - log(base 3)5^4
Using the product rule of logarithms, we can rewrite the expression as:
log(base 3)(4^2/5^4)
Simplifying further, we have:
log(base 3)(16/625)
Now, let's rewrite the original equation:
log(base 3) x^2 = log(base 3)(16/625)
Since the logarithms have the same base, the equation can be rewritten as x^2 = 16/625.
To solve for x, we take the square root of both sides:
√(x^2) = √(16/625)
Simplifying further, we get:
|x| = 4/25
Since the absolute value of x can be positive or negative, we have two possible solutions:
1. x = 4/25
2. x = -4/25
Thus, the solutions for x are x = 4/25 and x = -4/25.
To solve for x in the equation log(base 3) x^2 = 2log(base 3)4 - 4log(base 3)5, we can start by simplifying the right side of the equation using the properties of logarithms.
First, we can apply the power rule of logarithms, which states that log(base b) a^n = n · log(base b) a. Applying this rule, we can write the equation as:
log(base 3) x^2 = log(base 3) 4^2 - log(base 3) 5^4
Next, we can simplify the exponents:
log(base 3) x^2 = log(base 3) 16 - log(base 3) 625
Now, we can use the quotient rule of logarithms, which states that log(base b) (a / c) = log(base b) a - log(base b) c. Applying this rule, we can write the equation as:
log(base 3) x^2 = log(base 3) (16 / 625)
Now, we have a single logarithm on both sides of the equation. To eliminate the logarithm, we equate the arguments of the logarithm:
x^2 = 16 / 625
Next, we can simplify the right side of the equation:
x^2 = 0.0256
To solve for x, we take the square root of both sides:
x = ± √(0.0256)
Thus, the solutions for x are x = ± 0.16.