Question

Use the properties of logarithms to solve the equation for x. Remember to check your solution. ln (5x+3) = 18 (1 point) Responses x=215 x is equal to 21 fifths x=94 x is equal to 9 fourths x=(e18−3)5 x is equal to the fraction with numerator open paren e to the 18th power minus 3 close paren and denominator 5 x=(1018−3)5

Answer 1

To solve the equation \( \ln(5x + 3) = 18 \), we can use the properties of logarithms and exponentiation.

1. **Exponentiate both sides to eliminate the natural logarithm**:
\[
e^{\ln(5x + 3)} = e^{18}
\]
Simplifying this gives:
\[
5x + 3 = e^{18}
\]

2. **Isolate \( x \)**:
Subtract 3 from both sides:
\[
5x = e^{18} - 3
\]
Now divide by 5:
\[
x = \frac{e^{18} - 3}{5}
\]

This gives us one solution for \( x \):
\[
x = \frac{e^{18} - 3}{5}
\]

Now, let's check this solution by substituting \( x \) back into the original equation.

### Checking the solution:
1. Substitute back to the original equation:
\[
\ln(5 \left( \frac{e^{18} - 3}{5} \right) + 3)
\]
This simplifies to:
\[
\ln(e^{18} - 3 + 3) = \ln(e^{18})
\]
Since \( \ln(e^{18}) = 18 \), the left side matches the right side of the original equation.

### Conclusion:
The correct solution is:
\[
x = \frac{e^{18} - 3}{5}
\]

This corresponds to the response:
**x = (e^18 - 3)/5**