how do i solve these logs?
log1/2 (3x+1)^1/3= -2
3^(x^3)= 9^x
Wooah i don't get that ;]
do you understand mine?
Check your previous post for the solution to problem 2.
first problem
convert your equation to exponential form
(1/2)^-2 = (3x+1)^1/3
2^2 = (3x+1)^1/3
4 = (3x+1)^1/3 , cube both sides
64 = 3x+1
63 = 3x
x = 21
To solve equations involving logarithms, we need to work through some steps. Let's solve each equation step-by-step.
First, let's solve the equation: log₁/₂(3x+1)^(1/3) = -2.
Step 1: Isolate the logarithmic term.
To isolate the logarithmic term, we need to get rid of the base. In this case, the base is 1/2. We can do that by rewriting the equation in exponential form:
(3x+1)^(1/3) = (1/2)^(-2).
Step 2: Simplify the right side of the equation.
Recall that (1/2)^(-2) is equal to 2² = 4. So, the equation becomes:
(3x+1)^(1/3) = 4.
Step 3: Eliminate the exponent by raising both sides to the power of 3.
Cube both sides of the equation:
[(3x+1)^(1/3)]³ = 4³.
Simplifying, we have:
(3x+1) = 64.
Step 4: Solve for x.
Now, isolate x by subtracting 1 from both sides:
3x + 1 - 1 = 64 - 1,
3x = 63.
Divide both sides by 3:
3x/3 = 63/3,
x = 21.
Therefore, the solution to the equation log₁/₂(3x+1)^(1/3) = -2 is x = 21.
Now, let's move on to the second equation: 3^(x^3) = 9^x.
Step 1: Convert both sides to the same base.
To make the bases the same, we use the fact that 3 = 3², and 9 = 3^2. Rewriting the equation with a common base, we have:
(3²)^(x^3) = (3^2)^x.
Simplifying, we get:
3^(2x^3) = 3^(2x).
Step 2: Set the exponents equal to each other.
Since the bases are the same, we can set the exponents equal to each other:
2x^3 = 2x.
Step 3: Solve for x.
To solve this equation, we need to isolate x. Let's subtract 2x from both sides:
2x^3 - 2x = 0.
Step 4: Factor out x from both terms.
Factor out x:
x(2x^2 - 2) = 0.
Step 5: Solve for x by setting each factor equal to zero.
Option 1: x = 0.
Option 2: 2x^2 - 2 = 0.
For the second option, let's solve it separately:
2x^2 - 2 = 0.
Step 6: Isolate x by adding 2 to both sides:
2x^2 = 2.
Step 7: Divide both sides by 2:
x^2 = 1.
Step 8: Take the square root of both sides:
x = ±1.
Therefore, the solutions to the equation 3^(x^3) = 9^x are x = 0, and x = ±1.