Solve this logarithmic equation. 3 log 2 base x + log 18 base x = 2
To solve the logarithmic equation 3 log base x (2) + log base x (18) = 2, we can use the properties of logarithms to simplify the equation.
First, we can apply the change of base formula to convert the logarithms to a common base. Let's use logarithm base 10 for this example:
3 log base x (2) = log base 10 (2)
log base x (18) = log base 10 (18)
Applying the change of base formula, we have:
3 log base x (2) = log base 10 (2)
log base x (18) = log base 10 (18)
Now, let's use the property of logarithms that states:
a log base x (b) = log base x (b^a)
Using this property, we rewrite the equations as:
log base x (2^3) = log base 10 (2)
log base x (18) = log base 10 (18)
Simplifying these equations, we have:
log base x (8) = log base 10 (2)
log base x (18) = log base 10 (18)
Since the logarithms on both sides of the equations have the same base, we can equate the expressions inside the logarithms:
8 = 2
18 = 18
These equations are true, which means any value of x will make both sides equal. Therefore, x can be any positive number.
In conclusion, the solution to the logarithmic equation 3 log base x (2) + log base x (18) = 2 is x can be any positive number.