Exponential and Logarithmic Equations solve the equation. Find the exact solution if possible. otherwise, use...

Step 1
Law of logarithm:
Consider m to be a positive number and ${\displaystyle m\ne q1.}$
Again consider M and M to be any real numbers with ${\displaystyle M>0}$ and ${\displaystyle N>0.}$
The difference of logarithms of two numbers is equal to the logarithm of quotient of two numbers as,
${\displaystyle {\log }_{m}M-{\log }_{m}N={\log }_m\left(\frac{M}{N}\right)}$
The logarithm function with base m is denoted by ${\displaystyle {\log }_m}$ can be defined as,
${\displaystyle {\log }_{m}M=y}$
${\displaystyle M=m^y}$
Step 2
The given logarithm equation is,
1) ${\displaystyle {\log }_8(x+5)-{\log }_8(x-2)=1}$
The above logarithm equation can be combined from the laws of logarithm as,
2) ${\displaystyle {\log }_8\frac{(x+5)}{(x-2)}=1}$
The equation (2) can be expressed as,
${\displaystyle \frac{(x+5)}{(x-2)}=8^1}$
${\displaystyle x+5=8(x-2)}$
${\displaystyle x+5=8x-16}$
${\displaystyle x-8x=-16-5}$
Simplify above equation as,
${\displaystyle -7x=-21}$
${\displaystyle 7x=21}$
${\displaystyle x=\frac{21}{7}}$
${\displaystyle x=3}$
Therefore, ${\displaystyle x=3}$ is the solution of equation ${\displaystyle {\log }_8(x+5)-{\log }_8(x-2)=1.}$
Conclusion:
Thus, the solution of the logarithm equation ${\displaystyle {\log }_8(x+5)-{\log }_8(x-2)=1}$ is 3.