Let's solve the equation [tex]\( \log_3(z - x) = 3 \)[/tex] step by step.
### Step 1: Understand the equation
The equation given is:
[tex]\[ \log_3(z - x) = 3 \][/tex]
This reads as "the logarithm to the base [tex]\(3\)[/tex] of [tex]\(z - x\)[/tex] is [tex]\(3\)[/tex]."
### Step 2: Convert the logarithmic form to exponential form
To make this easier to solve, we'll convert the logarithmic equation into its exponential form. Recall that:
[tex]\[ \log_b(a) = c \iff a = b^c \][/tex]
Here, [tex]\(b = 3\)[/tex], [tex]\(a = z - x\)[/tex], and [tex]\(c = 3\)[/tex]. Therefore, we can write:
[tex]\[ z - x = 3^3 \][/tex]
### Step 3: Calculate the exponential term
Next, we'll calculate [tex]\(3^3\)[/tex]:
[tex]\[ 3^3 = 3 \times 3 \times 3 = 27 \][/tex]
So, the equation becomes:
[tex]\[ z - x = 27 \][/tex]
### Step 4: Solve for [tex]\(z\)[/tex]
Now we have a simple linear equation:
[tex]\[ z - x = 27 \][/tex]
To isolate [tex]\(z\)[/tex], we'll add [tex]\(x\)[/tex] to both sides of the equation:
[tex]\[ z = x + 27 \][/tex]
### Conclusion
The solution for [tex]\(z\)[/tex] in terms of [tex]\(x\)[/tex] is:
[tex]\[ z = x + 27 \][/tex]
This is the necessary solution to the equation [tex]\(\log_3(z - x) = 3\)[/tex].
