Understanding the Logarithmic Equation

We are asked to solve logarithmic equation: $\log_{10}(x^2 - 6x + 45) = 2$. This is a common type of math problem that involves converting a logarithm into an exponential form to make it easier to solve. To find the value of x, we need to eliminate the logarithm.

Converting from Logarithmic to Exponential Form

The definition of a logarithm states that $\log_b a = c$ is equivalent to $b^c = a$. In our equation, the base is $b = 10$, the argument is $a = x^2 - 6x + 45$, and the value of the logarithm is $c = 2$.

Applying the definition, we can rewrite the given logarithmic equation in exponential form:

$\log_{10}(x^2 - 6x + 45) = 2$ becomes $10^2 = x^2 - 6x + 45$.

Simplifying and Forming a Quadratic Equation

Next, we evaluate the exponential term:

$10^2 = 100$

So, the equation becomes:

$100 = x^2 - 6x + 45$

To solve logarithmic equation effectively, we need to rearrange this into the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$. We do this by moving all terms to one side of the equation:

$0 = x^2 - 6x + 45 - 100$

This simplifies to:

$x^2 - 6x - 55 = 0$

Now we have a standard quadratic equation that we need to solve for x.

Solving the Quadratic Equation

There are several methods to solve a quadratic equation, such as factoring, completing the square, or using the quadratic formula. Factoring is often the quickest method if it is possible. We need to find two numbers that multiply to $-55$ and add up to $-6$.

Let the two numbers be $p$ and $q$. We need $p \times q = -55$ and $p + q = -6$.

Let's list the factors of 55: (1, 55), (5, 11). Now consider the signs to get a product of -55 and a sum of -6.

• If $p = -11$ and $q = 5$: $p \times q = (-11) \times 5 = -55$ and $p + q = -11 + 5 = -6$. These are the numbers we need.

Now we can factor the quadratic equation:

$x^2 - 6x - 55 = 0$

$(x - 11)(x + 5) = 0$

To find the value of x, we set each factor equal to zero:

• $x - 11 = 0 \implies x = 11$
• $x + 5 = 0 \implies x = -5$

So, the possible values for x that satisfy the original logarithmic equation are $11$ and $-5$. We successfully used algebra to transform and solve logarithmic equation.

Conclusion

The values of x that satisfy the equation $\log_{10}(x^2 - 6x + 45) = 2$ are $11$ and $-5$. This process involved using the definition of a base 10 logarithm and equation solving techniques for quadratic equations.