To solve the equation [tex]\(\log_2(3x - 7) = 3\)[/tex], let's follow these steps:
1. Understand the logarithmic equation: The given equation is in logarithmic form, which can be expressed as:
[tex]\[
\log_2(3x - 7) = 3
\][/tex]
This means that the expression [tex]\(3x - 7\)[/tex] is the power to which 2 must be raised to get the number 8.
2. Rewrite the logarithmic equation in exponential form: The equation [tex]\(\log_2(3x - 7) = 3\)[/tex] can be rewritten as:
[tex]\[
3x - 7 = 2^3
\][/tex]
Here, [tex]\(2^3\)[/tex] represents 2 raised to the power of 3, which equals 8.
3. Simplify the exponential equation: Now, simplify the equation by substituting [tex]\(2^3\)[/tex] with 8:
[tex]\[
3x - 7 = 8
\][/tex]
4. Solve for [tex]\(x\)[/tex]: To isolate [tex]\(x\)[/tex], first add 7 to both sides of the equation:
[tex]\[
3x = 8 + 7
\][/tex]
This simplifies to:
[tex]\[
3x = 15
\][/tex]
5. Divide by 3: Finally, divide both sides by 3 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{15}{3}
\][/tex]
Simplifying the fraction gives:
[tex]\[
x = 5
\][/tex]
So, the solution to the equation [tex]\(\log_2(3x - 7) = 3\)[/tex] is [tex]\(x = 5\)[/tex].
