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Solve the equation. Write the solution set with the exact solutions.

[tex]\[ 3 \log _3(5w+3) + 5 = 20 \][/tex]

If there is more than one solution, separate the answers with commas. If there is no solution, write \{\}.

The exact solution set is [tex]$\square$[/tex].


Sagot :

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].