Solve this logarithmic equation. 3 log 2 base x + log 18 base x = 2
To solve the logarithmic equation 3 log2(x) + log18(x) = 2, we can use the properties of logarithms to simplify the equation before finding its solution.
Let's start by using the logarithmic property log a + log b = log(ab) to combine the two logarithms on the left side of the equation:
3 log2(x) + log18(x) = 2
log2(x^3) + log18(x) = 2
log2(x^3) + log2(18)/log2(x) = 2
log2(x^3) + log2(9 * 2)/log2(x) = 2
log2(x^3) + (log2(9) + log2(2))/log2(x) = 2
log2(x^3) + (log2(3^2) + log2(2))/log2(x) = 2
log2(x^3) + (2 log2(3) + log2(2))/log2(x) = 2
Since the bases of the logarithms are the same, we can combine the terms inside the parentheses:
log2(x^3 * 3^2 * 2) = 2
log2(18x^3) = 2
Now we can rewrite the equation as:
2 = log2(18x^3)
To get rid of the logarithm, we can rewrite the equation in exponential form:
2 = log2(18x^3)
2 = 18x^3
Now solve for x by isolating the variable:
18x^3 = 2
x^3 = 2/18
x^3 = 1/9
To find the cube root of both sides, we get:
x = ∛(1/9)
Therefore, the solution to the logarithmic equation 3 log2(x) + log18(x) = 2 is x = ∛(1/9).