TLDR: The answer is -2.
What you have here is a function with a point on the graph. It is understood that “x” represents the input and “y” represents the output, so we’re looking for the right “x” that will give y = 1/100.
To start, we can substitute the y-value into the function like this:
y = 10^x
1/100 = 10^x
From here, we can take two different pathways; one requires conceptual knowledge while the other relies on the knowledge of logarithms.
Conceptual
For the conceptual, we know that 100 = 10^2, so 1/100 is the same as 1/10^2. We also know that the inverse of a fraction is the same as the number to the negative of its power (for example, 1/2 is equal to 2^-1), so we know that 1/10^2 is the same as 10^-2. So far, we have learned that:
1/100 = 10^x
1/10^2 = 10^x
10^-2 = 10^x
So, to satisfy this, “x” must equal -2.
Logarithms
For logarithms, we can use powers and math to actually calculate the value of “x”. We know that:
1/100 = 10^x
To solve this, we need “x” to be by itself on one side of the equation. To do this, we can perform the inverse function of an exponent, which is a logarithm. In this case, the base of 10^x is 10, so we need a “base 10” logarithm to solve this function. Apply this function to both sides and simplify:
1/100 = 10^x
log10(1/100) = log10(10^x)
log10(1/100) = x
The log10 function is the inverse of 10^x, so they cancel out to leave “x”. Plug log10(1/100) into a calculator, and you find that x = -2.
Hope this helps!
