To solve the equation [tex]\(\log_7(x - 4) = \log_7(4x + 5)\)[/tex], let's go through the following steps:
1. Recognize that the logarithms are equal:
Since the logarithms with the same base are equal, the arguments must be equal as well. This implies:
[tex]\[
x - 4 = 4x + 5
\][/tex]
2. Isolate the variable [tex]\(x\)[/tex]:
To solve for [tex]\(x\)[/tex], first get all [tex]\(x\)[/tex]-terms on one side and constants on the other:
[tex]\[
x - 4 - 4x = 5
\][/tex]
Simplify the equation:
[tex]\[
-3x - 4 = 5
\][/tex]
Add 4 to both sides:
[tex]\[
-3x = 9
\][/tex]
Finally, divide both sides by -3:
[tex]\[
x = -3
\][/tex]
3. Check the solution:
Substitute [tex]\(x = -3\)[/tex] back into the original expressions to ensure it lies within the domains:
[tex]\[
x - 4 = -3 - 4 = -7
\][/tex]
[tex]\[
4x + 5 = 4(-3) + 5 = -12 + 5 = -7
\][/tex]
The domain of the logarithmic function requires its argument to be positive. Since [tex]\(-7\)[/tex] is not positive, at [tex]\(x = -3\)[/tex], the logarithmic expression is undefined.
Given these points, after solving the equation and verifying whether the resulting value fits the domain of the logarithmic functions, we conclude that despite the mathematical solution pointing to [tex]\(-3\)[/tex], it does not lie within the domain. Therefore:
There is no solution.
