1. logaB = 2 and logbC = 4, evaluate logcA
*the lowercase letters are bases ie. log(base a)B
2. solve for x: (log(base 2)x))^3 = log(base 2)x
To evaluate logcA, we need to express A in terms of C. From the given information, we have logaB = 2 and logbC = 4.
To find logcA, we need to express A in terms of C using the change of base formula: logaB = logbB / logba.
Let's start by expressing B in terms of A using the change of base formula, substituting logaB = 2:
2 = logbB / logba
Since logbB = 1, we have:
2 = 1 / logba
Rearranging the equation, we get:
logba = 1/2
Now, let's express C in terms of A using the change of base formula, substituting logbC = 4:
4 = logcC / logbc
Since logcC = 1, we have:
4 = 1 / logbc
Rearranging the equation, we get:
logbc = 1/4
Finally, we can combine the expressions for logba and logbc to find logcA:
logcA = logba / logbc
Substituting the given values:
logcA = (1/2) / (1/4)
Simplifying the expression, we get:
logcA = (1/2) * (4/1) = 2
Therefore, logcA = 2.
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To solve for x in the equation (log(base 2)x))^3 = log(base 2)x, we can follow these steps:
1. Rewrite the equation as a single logarithm:
(log(base 2)x)^3 = log(base 2)x
2. Simplify the equation by applying the power rule of logarithms:
log(base 2)(x^3) = log(base 2)x
3. Using the definition of logarithms, set the expressions inside the logarithms equal to each other:
x^3 = x
4. Rearrange the equation to solve for x:
x^3 - x = 0
5. Factor out x from the equation:
x(x^2 - 1) = 0
6. Apply the zero product property, which states that if a product of factors is equal to zero, then at least one of the factors must be zero:
x = 0 or x^2 - 1 = 0
7. Solve each equation separately:
If x = 0, then it satisfies the equation.
If x^2 - 1 = 0, then we can factor it as (x - 1)(x + 1) = 0. Applying the zero product property again, we have:
x - 1 = 0 or x + 1 = 0
Solving these equations:
If x - 1 = 0, then x = 1.
If x + 1 = 0, then x = -1.
Therefore, the solutions to the equation (log(base 2)x))^3 = log(base 2)x are x = 0, x = 1, and x = -1.