3 \log_{ 10 }({ 2x }) = 4

To solve the equation, we can start by using the property of logarithms that states:

n log_b(x) = log_b(x^n)

Therefore, we can rewrite the equation as:

log_{10}((2x)^3) = 4

Applying the power rule of logarithms, we can simplify further:

log_{10}(8x^3) = 4

Now, we can rewrite the equation in exponential form:

10^4 = 8x^3

Simplifying 10^4 gives us:

10000 = 8x^3

Dividing both sides of the equation by 8:

1250 = x^3

To find the value of x, we take the cube root of both sides:

x = ∛1250

Simplifying the cube root gives us:

x = 10∛5

Therefore, the solution for the equation is x = 10∛5.