I am in grade 10, but my question does not relate to the curriculum.
I am a student asking the question.
If x = y^z is there a way to solve for z, if x and y are given, without guessing or already knowing the answer?
ie: 64 = 2^z
6 is the obvious value for z. Is there a way I could solve for this variable using a formula, which would also apply when more complex values were substituted for x and y?

Nathan

Hi Nathan,

As you have noticed, solving x = y^z for z is not always obvious. What you need is a logarithm function. You will probably learn about logarithms in grade 12. The common logarithm, or sometimes called "log base 10" is the one that is most often used. It is written log(x) or log₁₀(x), and is defined for positive values of x by

y = log₁₀(x) if x = 10^y

If you have a scientific calculator it will have a logarithm function, perhaps even two logarithm functions. You can check that you are using the common (base 10) log by finding log(10) since log(10)=1 (can you see from the definition why this is true?)

The reason that the log function solves your problem is the fact that

log(a^b) = b log(a)

Here is how I would use logarithms to solve 64 = 2^y for y.

64 = 2^y, so

log(64) = log(2^y) = y log(2)

I then use my calculator to get

1.80617 = y 0.30103, so

y = 1.80617/0.30103 = 6.0

Using your calculator you get a decimal approximation where you knew that the answer is exactly 6, however this method will work even when the answer is not an integer, for example to find y if 63 = 2^y.

Cheers,
Penny

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