To determine which logarithmic equation matches the solution of the equation [tex]\(x - 4 = 2^3\)[/tex], let us follow a detailed step-by-step process:
1. First, consider the given equation [tex]\(x - 4 = 2^3\)[/tex].
2. Evaluate the right-hand side:
[tex]\[
2^3 = 8
\][/tex]
3. The equation now becomes:
[tex]\[
x - 4 = 8
\][/tex]
4. To solve for [tex]\(x\)[/tex], add 4 to both sides of the equation:
[tex]\[
x = 8 + 4
\][/tex]
[tex]\[
x = 12
\][/tex]
Thus, [tex]\(x = 12\)[/tex] is the solution to the equation [tex]\(x - 4 = 2^3\)[/tex].
Now, let's find the logarithmic equation that has the same solution as [tex]\(x - 4 = 2^3\)[/tex]:
To match the solution to the logarithmic form, note that the equation can also be written as an exponential function in the form of logarithms. Observe the options provided:
1. [tex]\(\log 3^2 = (x - 4)\)[/tex]
2. [tex]\(\log 2^3 = (x - 4)\)[/tex]
3. [tex]\(\log _2(x - 4) = 3\)[/tex]
4. [tex]\(\log _3(x - 4) = 2\)[/tex]
Consider each option to see which one corresponds to the original equation:
- Option 1: [tex]\(\log 3^2 = (x - 4)\)[/tex]
This simplifies to:
[tex]\[
\log 9 = (x - 4)
\][/tex]
To check if [tex]\(x = 12\)[/tex] fits:
[tex]\[
\log 9 = (12 - 4)
\][/tex]
[tex]\[
\log 9 \neq 8
\][/tex]
Therefore, this is not the correct equation.
- Option 2: [tex]\(\log 2^3 = (x - 4)\)[/tex]
This simplifies to:
[tex]\[
\log 8 = (x - 4)
\][/tex]
To check if [tex]\(x = 12\)[/tex] fits:
[tex]\[
\log 8 = (12 - 4)
\][/tex]
[tex]\[
\log 8 \neq 8
\][/tex]
Therefore, this is not the correct equation.
- Option 3: [tex]\(\log _2(x - 4) = 3\)[/tex]
Using the solution [tex]\(x = 12\)[/tex]:
[tex]\[
\log _2(12 - 4) = 3
\][/tex]
[tex]\[
\log _2(8) = 3
\][/tex]
We know that:
[tex]\[
2^3 = 8
\][/tex]
Therefore, [tex]\(\log _2 8 = 3\)[/tex], which is true. This matches the given solution.
- Option 4: [tex]\(\log _3(x - 4) = 2\)[/tex]
Using the solution [tex]\(x = 12\)[/tex]:
[tex]\[
\log _3(12 - 4) = 2
\][/tex]
[tex]\[
\log _3(8) = 2
\][/tex]
We know that:
[tex]\[
3^2 = 9 \implies \log _3 9 = 2
\][/tex]
However, [tex]\(8 \neq 9\)[/tex], so this is not the correct equation.
Thus, the logarithmic equation that has the same solution as [tex]\(x - 4 = 2^3\)[/tex] is:
[tex]\[
\boxed{\log_2(x - 4) = 3}
\][/tex]
