Video Transcript
Determine the number whose logarithm with base square root of five is five.
In this question, we’re asked to determine the value of a number. Let’s call this number 𝑥. We’re told that the logarithm with base square root of five of this number 𝑥 is equal to five. In other words, 𝑥 is a solution to the equation the log base square root of five of 𝑥 is equal to five. We can determine the value of 𝑥 by recalling how we could find the logarithms function. We recall the logarithmic functions are defined to be the inverse of exponential functions. In particular, if the log base 𝑏 of 𝑥 is equal to 𝑦, then we can conclude that 𝑥 is equal to 𝑏 to the power of 𝑦. And it is worth noting this is only true provided 𝑏 and 𝑥 are positive numbers and 𝑏 is not equal to one.
In our case, the value of 𝑏 is root five, the value of 𝑦 is five, and 𝑥 is just equal to itself. So applying this, we have 𝑥 is equal to root five raised to the fifth power. There’s then a few different ways of evaluating this. We could use our laws of exponents to write five as five to the power of one-half. And then, raising this all to the power of five means we multiply the exponent. That’s five to the power of five over two. However, in this case, since our exponent is a positive integer, we can just write root five to the fifth power as a product of root five, where this appears five times in the product.
We can then simplify this by noting root five times root five is equal to five, giving us five times five times root five, which simplifies to give us 25 root five, which is our final answer. Therefore, we were able to show the number whose logarithm with base root five is five is 25 root five.