QuestionSolve the equation: $\log _{3} x + \log _{3}(x-24) = 4$

Studdy Solution

STEP 1

Assumptions1. We are using the logarithmic properties to solve the equation. . The base of the logarithm is3.
3. The equation is $\log{3} x+\log{3}(x-24)=4$ .

STEP 2

We can use the property of logarithms that states the sum of two logarithms with the same base is the logarithm of the product of the two numbers. This allows us to combine the two logarithms on the left side of the equation.
$\log{} x+\log{}(x-24) = \log{}(x \cdot (x-24))$

STEP 3

Now, we can rewrite the equation as $\log{3}(x \cdot (x-24)) =$

STEP 4

We can convert the logarithmic equation to an exponential equation using the definition of logarithms. The base of the logarithm becomes the base of the exponent, the right side of the equation becomes the exponent, and the argument of the logarithm becomes the result.
$x \cdot (x-24) =3^4$

STEP 5

Calculate the right side of the equation.
$x \cdot (x-24) =81$

STEP 6

Now, we can distribute the x on the left side of the equation.
$x^2 -24x =81$

STEP 7

We can rewrite the equation in standard quadratic form, $ax^2 + bx + c =0$ , by subtracting81 from both sides of the equation.
$x^2 -24x -81 =0$

STEP 8

Now, we can factor the quadratic equation.
$(x -27)(x +3) =0$

STEP 9

Set each factor equal to zero and solve for x.
$x -27 = \quad \text{or} \quad x +3 =$

STEP 10

olving the equations gives $x =27 \quad \text{or} \quad x = -3$

STEP 11

However, since we cannot take the logarithm of a negative number, $x = -3$ is not a valid solution. Therefore, the only solution to the equation is $x =27$ .
The solution to the equation is $x =27$ .

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