8 = 2³

(2³)^x = 2^3x

2^3x = 2^x+6  <- when the bases are the same, you can set the exponents equal to each other and solve for x

3x=x+6

2x=6

x=3

First rewrite the two expressions so that they have the same base, and then compare their exponents.

\(\displaystyle 27 = 3^{3}\)   

\(\displaystyle 27^2 = (3^{3})^2\)

Combine exponents by multiplying: \(\displaystyle (3^{3})^2 = 3^6\)

This is the same as the first given expression, so the two expressions are equal.

\(\displaystyle 32\) can be written as \(\displaystyle 2^5.\) 

Since there is a common base of \(\displaystyle 2\), we can say

 \(\displaystyle 2^x=2^5\) or \(\displaystyle x=5\). 

The basees don't match.

However: 

\(\displaystyle \frac{1}{3}=3^{-1}\) thus we can rewrite the expression as \(\displaystyle \frac{1}{9}=\left(\frac{1}{3}\right)^2\).

Anything raised to negative power means \(\displaystyle 1\) over the base raised to the postive exponent. 

So, \(\displaystyle (3^{-1})^{2}\)\(\displaystyle =3^{-2}\). \(\displaystyle x=-2\). 

The bases don't match.

However: 

\(\displaystyle 4=\frac{1}{4}^{-1}\) and we recognize that \(\displaystyle 256=4^4\).

Anything raised to negative power means \(\displaystyle 1\) over the base raised to the postive exponent. 

\(\displaystyle \left(\frac{1}{4}^{-1}\right)^4=\frac{1}{4}^{-4}\).  

 \(\displaystyle x=-4\)

Recall that \(\displaystyle 128=2^7\). 

With same base, we can write this equation: 

\(\displaystyle x+1=7\). 

By subtracting \(\displaystyle 1\) on both sides, \(\displaystyle x=6\). 

Since \(\displaystyle 32=2^5\) we can rewrite the expression.

With same base, let's set up an equation of \(\displaystyle x^2+4=5\).

By subtracting \(\displaystyle 4\) on both sides, we get \(\displaystyle x^2=1\).

Take the square root of both sides we get BOTH \(\displaystyle 1\) and \(\displaystyle -1\). 

They don't have the same base, however: \(\displaystyle 25=5^2\).

Then \(\displaystyle 25^4=(5^2)^4\). You would multiply the \(\displaystyle 2\) and the \(\displaystyle 4\) instead of adding.

\(\displaystyle 2\cdot 4=8\). 

\(\displaystyle x=8\)

There are two ways to go about this.

Method \(\displaystyle 1:\)

They don't have the same bases however: \(\displaystyle 16=4^2\). Then \(\displaystyle 16^6=(4^2)^6\)

You would multiply the \(\displaystyle 2\) and the \(\displaystyle 6\) instead of adding. We have \(\displaystyle 2\cdot 6=2x\)

Divide \(\displaystyle 2\) on both sides to get \(\displaystyle x=6\).

Method \(\displaystyle 2\):

We can change the base from \(\displaystyle 4\) to \(\displaystyle 16.\)

\(\displaystyle 4^{2x}=(4^2)^x=16^x\) 

This is the basic property of the product of power exponents. 

We have the same base so basically \(\displaystyle x=6\). 

Since we can write \(\displaystyle 1024^x=(2^{10})^x\). 

With same base we can set up an equation of \(\displaystyle 10x=1\) 

Divide both sides by \(\displaystyle 10\) and we get \(\displaystyle x=\frac{1}{10}\).