Question

solve the equation log2 (x^2-12x+45)=3

Answer 1

To solve the equation \( \log_2 (x^2 - 12x + 45) = 3 \), we can first rewrite the logarithmic equation in exponential form. Recall that if \( \log_b(a) = c \), then \( a = b^c \).

Thus, we have:

\[
x^2 - 12x + 45 = 2^3
\]

Calculating \( 2^3 \):

\[
2^3 = 8
\]

Now, we can set up the equation:

\[
x^2 - 12x + 45 = 8
\]

Next, we subtract 8 from both sides:

\[
x^2 - 12x + 45 - 8 = 0
\]
\[
x^2 - 12x + 37 = 0
\]

Now, we will use the quadratic formula to solve for \( x \). The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In our equation, \( a = 1 \), \( b = -12 \), and \( c = 37 \).

Now we calculate the discriminant:

\[
b^2 - 4ac = (-12)^2 - 4(1)(37) = 144 - 148 = -4
\]

Since the discriminant is negative (\(-4\)), the equation has no real solutions. Therefore, there are no real values of \( x \) that satisfy the original equation \( \log_2 (x^2 - 12x + 45) = 3 \).

In conclusion, the result is:

\[
\text{No real solutions.}
\]