Give an example of a logarithmic equation that cannot be solved and explain why it cannot be solved.
work backwards
You know that we cannot take the log of a negative number, so
let x = -5
2x = x-5
take log of both sides
log 2x = log ( x-5)
now try to solve it
Ahh I see thx
An example of a logarithmic equation that cannot be solved is:
log(x) + log(x - 5) = 0
This equation cannot be solved because the sum of logarithms is only defined when both logarithms have the same base and the arguments are positive. However, in this case, we have two logarithms with base 10, which is the default base, and the argument for the second logarithm (x - 5) can potentially be negative.
For this equation to have a solution, both arguments of the logarithms must be positive. So, for log(x) to be defined, x must be greater than 0. Also, for log(x - 5) to be defined, x - 5 must be greater than 0. This leads to the condition that x > 5.
Therefore, the equation log(x) + log(x - 5) = 0 cannot be solved as there is no single value of x that satisfies both conditions x > 0 and x > 5 simultanously.
A logarithmic equation that cannot be solved is one in which the variable is inside the logarithm function and also appears in other terms or functions in the equation. Let's take an example:
log(x) + x = 5
In this equation, the variable 'x' appears both inside the logarithm function (log(x)) and as a separate term (x). This makes it challenging to solve analytically because there is no straightforward algebraic manipulation to isolate 'x'.
Logarithmic functions and other algebraic expressions do not interact with each other in simple ways, making it difficult to find a direct solution. Though numerical methods, such as trial and error or approximation techniques, can be used to estimate a solution, there is no algebraic method to find an exact solution. Hence, the equation cannot be solved explicitly using conventional mathematical techniques.