Question

The number of customers at a new taco truck triples every day for a week. The function f(x) = 3^x represents the number of customers on day x. On what day was there 243 customers? Write an equation you need to solve it

Answer 1

Day 5

Explanation:

There are 2 ways to solve this, one using exponents and the other using logs. The equation regardless of how you solve it looks like this:

`y=3^x`

where y is the number of customers on day x. Filling in the number of customers we were given:

`243=3^x`

There's a rule in exponents that says if the bases on both sides of the equals sign are like, then their exponents are equal to each other. So if we can rewrite 243 in terms of a base of 3, then we're good. I went to my calculator and started raising 3 to increasing powers of x: 3 to the first is 3; 3 squared is 9; 3 cubed is 27; 3 to the fourth is 81; 3 to the fifth is 243! That means that

`3^5=3^x`

Since the bases are like, then x = 5.

Using logs to solve it:

Take the log of both sides:

`log(243)=log(3)^x`

There's a law of logs that says when you take the log of a number with an exponent, you can bring down the exponent in front like this:

`log(243)=xlog(3)`

Now to solve for x, divide both sides by log(3):

`(log(243))/(log(3))=x`

You can do that on your calculator and get that x = 5.

Whichever way is easier for you to understand OR whichever way your teacher is teaching according to where you are in your log unit, do it that way! You can see that both give the same answer, the methods to get there vary.