How do I reverse engineer this "power of"/exponent?

Question

Take the following:

(2)^3 = 8

I understand that this is

2 * 2 * 2 = 8

My question is how do I reverse engineer this if I do not know the power like this:

(2)^x = 8

What is the value of x?

x could potentially contain a decimal and so could the result:

(2)^1.5 = 2.82842712474619

So without any numbers it would be:

(y)^x = z

How do I find out what x is?

x = ?

Answer 1

The short answer is "take logs". The logarithm, or log, of a number reflects what power you need to raise a certain base to in order to get that number.

For interesting but complicated reasons, the number $e=2.718281828...$ is often used as the base, in which case the "log" function is written "$\ln$" ("natural logarithm"), but any other base $b$ will work the same way ("$\log_b$") for the equation you're interested in.

The reason this works is that $\ln y^x = x\ln y$, so

$$x=\frac{\ln z}{\ln y}$$

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(The reason this is consistent with @amWhy's answer is because $\frac{\log_a b}{\log_a c}=\log_c b$.)

Answer 2

See logarithm. That's essentially the inverse function of an exponential function. In your case $$2^x = 8 \iff x = \log_2(8)$$

In general: $$a^x = b \iff x = \log_a(b)$$