Question #ec244

2 Answers
Oct 12, 2017

53x=127

Explanation:

We know that 5x=3

Remember the rule of exponents:

(ab)c=abc

Therefore:

(5x)3=5(x)(3)=53x

So we need to raise 5x to the "-3"rd power to get 53x.

(5x)3=33=133=127

Final Answer

Oct 13, 2017

53(log(3)log(5))
Or 0.37037....

Explanation:

In order to solve this, you can use a calculator or just use log.

However....if you don't know what log is, then allow me to give you a quick demonstration.
logaab=b

This might seem a little confusing at first, but the a is the "base". B is supposed to be the exponent of a in order to get ab.

An actual example: log525=2.

As you could see, 5 is the a in this situation and b is the exponent therefore the "middle number" is 25 (because 52=25).

Rule: If there is no base or a, then the a value is automatically assumed to be 10.

Now, another essential rule is that if a^b has an actual exponent on it (such as log1002), we can move that exponent to the left of log so it multiplies the result of log 100.

Proof: log1002 or log10000.
log10000=4 (because 104=10000).

Now, let's multiply the result by two...

2log100

log100=2

Let's multiply the result by two...

22=4! Therefore it works!

As you can see here,

Firstly, log both sides in order to get:
log5x=log3.

Then, due to the rules of logarithms, we can move the power of the x so it multiplies log5.

(x)log5=log3.

Divide both sides by log5

x=log3log5.

Then multiply both sides by negative one to get:

x=log3log5.

Now that we know what x is, we can plug it into the equation to get...

53(log3log5)

You can simplify the logs with a calculator in order to get 0.37037....

Forgive me for this long answer.