find x from log3 x-3log3 x=2
if you let y = log3x, you have
y - 3y = 2
y = -1
so, log3 x = -1
x = 3^-1 = 1/3
I assume you mean:
log3 x-3log3 x=2
then
log3 x - log3 x^3 = 2
log3 (x/x^3) = 2
log3 (1/x^2) = 2
1/x^2 = 3^2 = 9
9x^2 = 1
x = ± 1/3, but log (-1/3) would be undefined, so
x = 1/3
To find the value of x from the equation log3(x) - 3log3(x) = 2, we can simplify the equation step by step:
Step 1: Combine the logarithms using logarithmic rules. Subtracting logarithms is equivalent to dividing the corresponding values inside the logarithms. Rewrite the equation as:
log3(x) / (log3(x))^3 = 2
Step 2: Simplify the denominator. (log3(x))^3 can be rewritten as (log3(x))^2 * log3(x). The equation becomes:
log3(x) / (log3(x))^2 * log3(x) = 2
Step 3: Simplify further. When dividing logarithms with the same base, we can subtract the exponents inside the logarithm. The equation can be rewritten as:
1 / (log3(x)) * log3(x) = 2
Step 4: Simplify the equation. The product of the logarithm and its inverse cancel each other out, resulting in:
1 = 2
Step 5: As there isn't a valid solution for the equation 1 = 2, we conclude that there is no real value of x that satisfies the original equation log3(x) - 3log3(x) = 2.
To find the value of x in the equation log3(x) - 3log3(x) = 2, we can use logarithmic properties and algebraic manipulation. Here's the step-by-step explanation:
Step 1: Apply the logarithmic properties.
Using the logarithmic property log3(a) - log3(b) = log3(a/b), the equation can be rewritten as log3(x) - log3(x^3) = 2.
Step 2: Simplify the equation using the logarithmic property.
Using the property log3(a) - log3(b) = log3(a/b), the equation becomes log3(x/x^3) = 2.
Step 3: Simplify the expression inside the logarithm.
Simplifying the fraction x/x^3, we get 1/x^2. Therefore, the equation becomes log3(1/x^2) = 2.
Step 4: Rewrite the equation in exponential form.
In exponential form, log3(1/x^2) = 2 can be rewritten as 3^2 = 1/x^2.
Step 5: Solve for x.
Simplifying 3^2, we get 9, and the equation becomes 9 = 1/x^2.
Step 6: Solve for x^2.
To solve for x^2, isolate it on one side of the equation. Dividing both sides by 9, we get 1/9 = 1/x^2.
Step 7: Take the reciprocal of both sides.
Taking the reciprocal of both sides, we get 9/1 = x^2/1, which simplifies to x^2 = 9.
Step 8: Take the square root.
Taking the square root of both sides, we have √(x^2) = √(9), which simplifies to x = ±3.
Therefore, the values of x that satisfy the equation log3(x) - 3log3(x) = 2 are x = 3 and x = -3.