The solution to the logarithmic equation log(x + 1) = 5 log(x) - 2 is approximately x = 1.38028. The process involves simplifying the equation, setting logarithmic terms equal, and solving the resulting quintic equation. Note that x must be greater than 0 and not equal to 1.
To solve the logarithmic equation log(x + 1) = 5 log(x) - 2, simplify by combining logarithmic terms:
log(x + 1) = log(x^5) - 2
Using the property that equal logarithms imply equal arguments, set x + 1 equal to x^5:
x + 1 = x^5
Rearrange to get the quintic equation x^5 - x - 1 = 0. The solution, approximately x = 1.38028, can be found using numerical methods or calculators.
Note: x must be greater than 0 and not equal to 1, as the logarithm of 0 or 1 is undefined.
Complete question should be:
What is the solution to the logarithmic equation log(x + 1) = 5 log(x) - 2? Solve for x.