Determine the solution set of the equation log base four of 𝑥 plus 25 log base 𝑥 of four equals 10 in the set of real numbers.
In order to answer this question, we’ll use one of the change of base laws of logarithms. This states that log base 𝑎 of 𝑏 multiplied by log base 𝑏 of 𝑐 is equal to log base 𝑎 of 𝑐. This rule holds when 𝑎, 𝑏, and 𝑐 are positive numbers with 𝑎 and 𝑏 both not equal to one. Considering the two logarithmic terms in our equation, we see that log base four of 𝑥 multiplied by log base 𝑥 of four is equal to log base four of four. And since log base 𝑎 of 𝑎 equals one, this is also equal to one.
If we let 𝑦 equal log base four of 𝑥, then log base 𝑥 of four is equal to one over 𝑦. We can then rewrite the full equation as 𝑦 plus 25 multiplied by one over 𝑦 is equal to 10. Multiplying each term by 𝑦 gives us 𝑦 squared plus 25 is equal to 10𝑦. We can then subtract 10𝑦 from both sides of this equation, giving us a quadratic in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero.
We can solve the equation 𝑦 squared minus 10𝑦 plus 25 equals zero by factoring. This is equal to 𝑦 minus five multiplied by 𝑦 minus five, as negative five plus negative five equals negative 10 and negative five multiplied by negative five is 25. The quadratic equation has one solution: 𝑦 is equal to five.
Recalling that 𝑦 was equal to log base four of 𝑥, then log base four of 𝑥 must equal five. We know that since a logarithmic function is the inverse of an exponential function that if log base 𝑎 of 𝑥 equals 𝑏, then 𝑥 is equal to 𝑎 to the power of 𝑏. 𝑥 is therefore equal to four to the power of five or four to the fifth power. This is equal to 1,024.
The solution set of the equation log base four of 𝑥 plus 25 log base 𝑥 of four equals 10 is 1,024. There is only one real solution to this equation.