Solve for x:
log base5 (4x) + log base5 (x) = log base5 (64)
all logs base 5
64 is 4^3
log (4 x^2) = log 64 = log (4^3)
so
4 x^2 = 4^3
x = 4
To solve for x in the equation log base5 (4x) + log base5 (x) = log base5 (64), we can use the properties of logarithms to simplify the equation.
The first property we can use is the product property of logarithms, which states that the logarithm of a product is equal to the sum of the logarithms of the individual numbers. In other words, log base b (xy) = log base b (x) + log base b (y).
Using this property, we can rewrite the equation as:
log base5 (4x * x) = log base5 (64).
Next, we simplify the equation by multiplying the terms on the left side and simplifying the right side:
log base5 (4x^2) = log base5 (64).
Now, we can use the property of logarithms that states if log base b (x) = log base b (y), then x = y. This property allows us to equate the expressions inside the logarithms on both sides of the equation.
So, we have:
4x^2 = 64.
To solve for x, we can divide both sides of the equation by 4:
x^2 = 16.
Taking the square root of both sides gives us:
x = ±√(16).
Therefore, the solutions for x are x = 4 and x = -4.
To verify these solutions, substitute them back into the original equation and check if both sides are equal.