Find the value of 𝑥 for which log base 𝑥 of 243 is equal to negative five.
In order to answer this question, we recall that if log base 𝑎 of 𝑏 is equal to 𝑐, then 𝑎 to the power of 𝑐 is equal to 𝑏. In this question, we need to calculate the value of 𝑎. We know that 𝑏 is 243 and 𝑐 is equal to negative five. This means that log base 𝑥 of 243 equals negative five can be rewritten as 𝑥 to the power of negative five is equal to 243.
We know that when dealing with negative exponents or indices, 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛. This means that one over 𝑥 to the fifth power is equal to 243. We can multiply through by 𝑥 to the fifth power, giving us one is equal to 243 multiplied by 𝑥 to the fifth power. Dividing both sides of this equation by 243 gives us 𝑥 to the fifth power is equal to one over 243.
Finally, we can calculate the value of 𝑥 by taking the fifth root of one over 243. When taking the root of a fraction, we simply take the root of the numerator and the root of the denominator separately. The fifth root of one is equal to one and the fifth root of 243 is equal to three as three to the fifth power is 243.
The value of 𝑥 for which log to the base 𝑥 of 243 equals negative five is one-third.