# Question ec244

Oct 12, 2017

${5}^{3 x} = \frac{1}{27}$

#### Explanation:

We know that $\textcolor{red}{{5}^{-} x} = \textcolor{b l u e}{3}$

Remember the rule of exponents:

${\left({a}^{b}\right)}^{c} = {a}^{b c}$

Therefore:

${\left(\textcolor{red}{{5}^{-} x}\right)}^{- 3} = {5}^{\left(- x\right) \left(- 3\right)} = {5}^{3 x}$

So we need to raise ${5}^{- x}$ to the "-3"rd power to get ${5}^{3 x}$.

${\left(\textcolor{red}{{5}^{-} x}\right)}^{- 3} = {\textcolor{b l u e}{3}}^{- 3} = \frac{1}{\textcolor{b l u e}{3}} ^ 3 = \frac{1}{27}$

Oct 13, 2017

${5}^{3 \left(- \log \frac{3}{\log} \left(5\right)\right)}$
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.
${\log}_{a} {a}^{b} = b$

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 ${a}^{b}$.

An actual example: ${\log}_{5} 25 = 2$.

As you could see, 5 is the $a$ in this situation and $b$ is the exponent therefore the "middle number" is 25 (because ${5}^{2} = 25$).

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 $\log {100}^{2}$), we can move that exponent to the left of log so it multiplies the result of log 100.

Proof: $\log {100}^{2}$ or $\log 10000$.
$\log 10000 = 4$ (because ${10}^{4} = 10000$).

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

$2 \log 100$

$\log 100 = 2$

Let's multiply the result by two...

$2 \cdot 2 = 4$! Therefore it works!

As you can see here,

Firstly, $\log$ both sides in order to get:
$\log {5}^{-} x = \log 3$.

Then, due to the rules of logarithms, we can move the power of the $- x$ so it multiplies $\log 5$.

$\left(- x\right) \log 5 = \log 3$.

Divide both sides by $\log 5$

$- x = \log \frac{3}{\log} 5$.

Then multiply both sides by negative one to get:

$x = - \log \frac{3}{\log} 5$.

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

5^(-3(log3/log5)#

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

Forgive me for this long answer.