1. solve for x: (log(base 2)x))^3 = log(base 2)x
2. solve for x: log(base 3) x-1 = log(base 3) 18
To solve the first equation, (log(base 2)x)^3 = log(base 2)x, we can follow these steps:
Step 1: Simplify the equation by rewriting the cube on the left side.
(log(base 2)x) ⋅ (log(base 2)x) ⋅ (log(base 2)x) = log(base 2)x
Step 2: Since the base is the same on both sides of the equation (base 2), we can drop the logarithmic notation and write the equation as:
(log(base 2)x)^3 - log(base 2)x = 0
Step 3: Factor out the common term, log(base 2)x, from both terms:
(log(base 2)x) ⋅ [(log(base 2)x)^2 - 1] = 0
Step 4: Set each factor equal to 0 and solve for x separately:
Factor 1: (log(base 2)x) = 0
This gives us: x = 2^0 = 1
Factor 2: (log(base 2)x)^2 - 1 = 0
This gives us: (log(base 2)x)^2 = 1
Take the square root of both sides: log(base 2)x = ±1
Rewrite in exponential form: x = 2^1 = 2, or x = 2^-1 = 1/2
Therefore, the solutions to the equation (log(base 2)x)^3 = log(base 2)x are x = 1, 2, and 1/2.
Now let's move on to the second equation.
To solve the equation log(base 3) x-1 = log(base 3) 18, we can use the property of logarithms that states if the bases are the same, then the arguments of the logarithms are also the same. So, we have:
x-1 = 18
To solve for x, we add 1 to both sides:
x = 18 + 1 = 19
Therefore, the solution to the equation log(base 3) x-1 = log(base 3) 18 is x = 19.