11) You have the following equation:
[tex]\log _3(2x-7)=4[/tex]
In order to determine if the equation has solution. Proceed as follow:
- Use the equation above as the exponent of 3:
[tex]3^{\log _3(2x-7)}=3^4[/tex]
- Then, by properties of the logarithms, you can cancel out the log_3:
[tex]3^{\log _3(2x-7)}=2x-7=3^4[/tex]
- Now, consider the second and third member of the previous equation:
[tex]2x-7=3^4[/tex]
And solve for x by adding 7 both sides and then by dividing by 2 both sides:
[tex]\begin{gathered} 2x=3^4+7 \\ x=\frac{3^4+7}{2}=\frac{81+7}{2}=\frac{88}{2}=44 \end{gathered}[/tex]
Hence, the solution for x = 44
