Let's solve the equation [tex]\(\log_x(8x - 3) - \log_x 4 = 2\)[/tex] step by step.
1. Combine the logarithms:
Using the property of logarithms, [tex]\(\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)\)[/tex], we can combine the two logarithmic terms:
[tex]\[
\log_x\left(\frac{8x - 3}{4}\right) = 2
\][/tex]
2. Rewrite the logarithmic equation as an exponential equation:
Recall that if [tex]\(\log_b(y) = z\)[/tex], then [tex]\(b^z = y\)[/tex]. Applying this property, we get:
[tex]\[
x^2 = \frac{8x - 3}{4}
\][/tex]
3. Clear the fraction:
Multiply both sides of the equation by 4:
[tex]\[
4x^2 = 8x - 3
\][/tex]
4. Form a standard quadratic equation:
Rearrange the equation:
[tex]\[
4x^2 - 8x + 3 = 0
\][/tex]
5. Solve the quadratic equation:
Use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex] for [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a = 4\)[/tex], [tex]\(b = -8\)[/tex], and [tex]\(c = 3\)[/tex]:
[tex]\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 4 \cdot 3}}{2 \cdot 4}
\][/tex]
Simplify the terms under the square root and solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{8 \pm \sqrt{64 - 48}}{8}
\][/tex]
[tex]\[
x = \frac{8 \pm \sqrt{16}}{8}
\][/tex]
[tex]\[
x = \frac{8 \pm 4}{8}
\][/tex]
6. Calculate the two possible values of [tex]\(x\)[/tex]:
[tex]\[
x = \frac{8 + 4}{8} = \frac{12}{8} = \frac{3}{2}
\][/tex]
[tex]\[
x = \frac{8 - 4}{8} = \frac{4}{8} = \frac{1}{2}
\][/tex]
Thus, the values of [tex]\(x\)[/tex] are [tex]\(\frac{3}{2}\)[/tex] and [tex]\(\frac{1}{2}\)[/tex], where [tex]\(\frac{3}{2}\)[/tex] is the greater value.
So, the value of [tex]\(x\)[/tex] is [tex]\(\frac{3}{2}\)[/tex] or [tex]\(\frac{1}{2}\)[/tex].
