log3(x^2)=2log3(4)-4log3(5)
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log3(x^2)=2log3(4)-4log3(5)
log3 x^2 = log34^2 - log35^4
log3x^2 = log3(16/625)
x^2 = 16/625
x = ± 4/25
To solve the equation log3(x^2) = 2log3(4) - 4log3(5), we can use the following properties of logarithms:
1. Product Rule: logb(x * y) = logb(x) + logb(y)
2. Power Rule: logb(x^a) = a * logb(x)
Let's simplify the equation step by step:
First, apply the Power Rule on the right side of the equation:
log3(x^2) = log3(4^2) - log3(5^4)
Simplify the right side:
log3(x^2) = log3(16) - log3(625)
Next, use the Product Rule on the right side by combining the logarithms:
log3(x^2) = log3(16/625)
Simplify the expression on the right side:
log3(x^2) = log3(16/625)
Now, since both sides of the equation have the same base (log3), we can drop the logarithm and focus on the argument:
x^2 = 16/625
To find x, we need to take the square root of both sides:
√(x^2) = √(16/625)
Simplify the square roots:
x = ±√(16/625)
Further simplify the right side fraction:
x = ±(4/25)
Therefore, the equation log3(x^2) = 2log3(4) - 4log3(5) has two solutions: x = 4/25 and x = -4/25.
To solve the equation log3(x^2) = 2log3(4) - 4log3(5), we'll need to apply logarithmic properties and simplify the equation step by step.
Step 1: Rewrite the equation using the power rule of logarithms.
log3(x^2) = log3(4^2) - log3(5^4)
Step 2: Simplify the right side of the equation.
log3(x^2) = log3(16) - log3(625)
Step 3: Simplify the logarithmic expressions on the right side using the quotient rule of logarithms.
log3(x^2) = log3(16 / 625)
Step 4: Simplify the fraction 16/625.
log3(x^2) = log3(0.0256)
Step 5: Set the logarithmic equation equal to each other.
x^2 = 0.0256
Step 6: Take the square root of both sides of the equation to isolate x.
x = ±√(0.0256)
Step 7: Simplify the square root.
x ≈ ±0.16
Therefore, the solutions for the equation log3(x^2) = 2log3(4) - 4log3(5) are approximately x = -0.16 and x = 0.16.