Combine, rewrite & solve:
4 log base 9 of 3 = x
log base x of 12 + log base x of 18 = 3
To solve the given equations, we'll need to combine, rewrite, and solve.
Equation 1: 4 * log base 9 of 3 = x
To combine and rewrite this equation, we'll use the property of logarithms that states that multiplying by a constant is equivalent to raising the base to that constant power. Therefore, we can rewrite equation 1 as:
log base 9 of 3^4 = x
Simplifying further:
log base 9 of 81 = x
Now, we'll solve for x by converting the logarithmic equation into exponential form. In exponential form, a logarithmic equation can be rewritten as:
9^x = 81
Since 81 can be expressed as 9^2, we have:
9^x = 9^2
By equating the bases, we can conclude that:
x = 2
Equation 2: log base x of 12 + log base x of 18 = 3
To combine these logarithms, we'll apply the logarithmic property that states that adding logarithms with the same base is equal to multiplying the arguments within the logarithms. Thus, we can rewrite equation 2 as:
log base x of (12 * 18) = 3
Simplifying within the logarithm:
log base x of 216 = 3
Next, we'll convert this logarithmic equation into exponential form:
x^3 = 216
Now, we can solve this equation by taking the cube root of both sides:
x = 6
Hence, the solution to the given equations is:
x = 2 (from equation 1: 4 * log base 9 of 3 = x)
x = 6 (from equation 2: log base x of 12 + log base x of 18 = 3)