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(Please use "^" to indicate an exponent. IOW: f(t) = 10^t)
with t as the time (in minutes) and f(t) is the number of viruses at time t. So to find when the number the viruses reaches 5 billion we replace f(t) with 5 billion and solve for t:
Solving for t will require that we get it out of the exponent. This is why we use logarithms. So we start by finding the logarithm of each side. You can use any base of logarithm but it will be easier if we use a base your calculator "knows" (like base 10 or base e (aka ln)). And since the equation has 10 to a power, it will be easiest if we use base 10. So that is what we will use:
There is a property of logarithms,
, which allows us to move an exponent of the argument out in front. (It is this property that is the reason we use logarithms to get variables out of exponents.) Using this property on your equation we get:
Since log(10) = 1 this simplifies to
This is an exact answer for the time when the number of viruses reaches 5 billion. If you want a decimal approximation for the answer, then get out your calculator and use it to find log(5000000000).