Certainly! Let's solve the given equation step by step:
We start with the equation:
[tex]\[
3 \log_3 (5w + 3) + 5 = 20
\][/tex]
1. Isolate the logarithmic term:
First, we need to isolate the logarithmic term by subtracting 5 from both sides of the equation:
[tex]\[
3 \log_3 (5w + 3) = 20 - 5
\][/tex]
This simplifies to:
[tex]\[
3 \log_3 (5w + 3) = 15
\][/tex]
2. Divide by the coefficient of the logarithmic term:
Next, we divide both sides by 3 to further isolate the logarithm:
[tex]\[
\log_3 (5w + 3) = \frac{15}{3}
\][/tex]
Simplifying the right-hand side, we get:
[tex]\[
\log_3 (5w + 3) = 5
\][/tex]
3. Solve the logarithm:
The equation [tex]\(\log_3 (5w + 3) = 5\)[/tex] means that [tex]\(5w + 3\)[/tex] is the number that 3 must be raised to the power of 5 to get. So, we can rewrite this in exponential form:
[tex]\[
5w + 3 = 3^5
\][/tex]
4. Calculate the exponential term:
Now, we calculate [tex]\(3^5\)[/tex]:
[tex]\[
3^5 = 243
\][/tex]
Thus, the equation becomes:
[tex]\[
5w + 3 = 243
\][/tex]
5. Solve for [tex]\(w\)[/tex]:
Finally, we solve for [tex]\(w\)[/tex] by isolating it:
[tex]\[
5w = 243 - 3
\][/tex]
Simplifying the right-hand side, we get:
[tex]\[
5w = 240
\][/tex]
Dividing both sides by 5, we obtain:
[tex]\[
w = \frac{240}{5} = 48
\][/tex]
Therefore, the exact solution set is [tex]\(\{48\}\)[/tex].
