Question

What is the solution to log^3(x+12) = log^3(5x)?

a) x = 2
b) x = 4
c) x = 6
d) x = 8

Answer 1

The solution to the equation
`\(\log^3(x+12) = \log^3(5x)\)` is (x = 4) (Option b).

Step-by-step explanation:

To solve the given logarithmic equation, we can start by taking the cube root of both sides, as the equation involves a cubic logarithm. This gives us
`\(\log(x+12) = \log(5x)\)`. Now, we can remove the logarithms by exponentiating both sides with the base 10, resulting in (x + 12 = 5x). Simplifying this linear equation, we find (x = 4), which is the solution.

Taking the cube root of both sides is a common technique when dealing with logarithmic equations with a cubic power. This step allows us to simplify the equation and remove the cube from the logarithm, making it more manageable. Exponentiating both sides with the base of the logarithm then allows us to obtain a simpler equation in terms of (x), which can be solved algebraically. In this case, (x = 4) satisfies the original logarithmic equation.

The solution (x = 4) is the only valid answer among the given options (a, b, c, d). It is essential to check potential solutions in the original equation to ensure they are valid, considering that logarithmic functions may have restrictions on their domains. In this case, (x = 4) is the correct solution that satisfies the given logarithmic equation.