Solve for x if (log_3x²)-1/2log_3x=3/2
To solve the equation, we first need to combine the logarithmic terms on the left side of the equation.
(log_3x²) - (1/2)log_3x = 3/2
We can use the logarithm rule that states: log_b(x^n) = n*log_b(x)
So, applying this rule to the equation, we have:
log_3(x²) - log_3(x^(1/2)) = 3/2
Now, we can use another logarithmic rule that says: log_b(x) - log_b(y) = log_b(x/y)
So, applying this rule to the equation, we have:
log_3((x²)/(x^(1/2))) = 3/2
Simplifying the expression inside the logarithm, we have:
log_3(x^(1.5)) = 3/2
Now, we can convert the equation to exponential form:
x^(1.5) = 3^(3/2)
Taking the square root of both sides:
√(x^(1.5)) = √(3^(3/2))
Simplifying, we have:
x^(0.75) = 3^(3/4)
Now, raise both sides to the power of 4/3:
(x^(0.75))^(4/3) = (3^(3/4))^(4/3)
Simplifying, we have:
x = 3^1
Therefore, the solution is x = 3.
To solve the equation (log_3(x^2)) - (1/2)log_3(x) = 3/2:
Step 1: Simplify the equation using logarithmic properties.
Using the property log_b(a^n) = n * log_b(a), we can rewrite the equation as:
2log_3(x) - (1/2)log_3(x) = 3/2
Step 2: Combine the like terms on the left side of the equation.
2log_3(x) - (1/2)log_3(x) = (3/2)
(4/2)log_3(x) - (1/2)log_3(x) = (3/2)
(3/2)log_3(x) = (3/2)
Step 3: Simplify the equation further.
The equation can be rewritten as:
2 * (3/2)log_3(x) = 3/2
3log_3(x) = 3/2
Step 4: Isolate the logarithmic term.
Divide both sides of the equation by 3:
log_3(x) = (3/2) / 3
log_3(x) = 1/2
Step 5: Convert the logarithmic equation to exponential form.
Using the base 3, rewrite the equation as:
x = 3^(1/2)
Step 6: Simplify the expression.
Since the square root of 3 is positive, x can be written as:
x = √3
Therefore, the solution to the equation (log_3(x^2)) - (1/2)log_3(x) = 3/2 is x = √3.