Begin by recognizing that both sides of the equation have a root term of $\displaystyle 3$.

$\displaystyle \small 9^x=3^6$

$\displaystyle (3^2)^x=3^6$

Using the power rule, we can set the exponents equal to each other.

$\displaystyle 3^{(2*x)}=3^6$

$\displaystyle \small 2x=6$

$\displaystyle \small x=3$

Begin by recognizing that both sides of the equation have the same root term, $\displaystyle 3$.

$\displaystyle \small 3^{2x}=81$

$\displaystyle \small 3^{2x}=9^2$

$\displaystyle 3^{2x}=(3^2)^2$

We can use the power rule to combine exponents.

$\displaystyle 3^{2x}=3^4$

Set the exponents equal to each other.

$\displaystyle 2x=4$

$\displaystyle \small x=2$

Solve for the values of a and b:

In 2009, $\displaystyle y=1034$ and $\displaystyle x=0$ (zero years since 2009). Plug this into the exponential equation form:

$\displaystyle 1034=ab^{(0)}$. Solve for $\displaystyle a$ to get  $\displaystyle a=1034$.

In 2013, $\displaystyle y=1711$ and $\displaystyle x=4$. Therefore,

$\displaystyle 1711=ab^4$  or  $\displaystyle 1711=(1034)b^4$.   Solve for $\displaystyle b$ to get

$\displaystyle b=\sqrt[4]{\frac{1711}{1034}}\approx1.1342$.

Then the exponential growth function is  

$\displaystyle y=1034(1.1342)^{x}$.

8 and 4 are both powers of 2.

$\displaystyle 4^{5x}=8^{4x-1}$

$\displaystyle 2^{10x}=2^{12x-3}$

$\displaystyle 10x=12x-3 -2x=-3 x=1.5$

Because both sides of the equation have the same base, set the terms equal to each other.

$\displaystyle 2x+3=5x-9$

Add 9 to both sides: $\displaystyle 2x+12=5x$

Then, subtract 2x from both sides: $\displaystyle 12=3x$

Finally, divide both sides by 3: $\displaystyle x=4$

125 and 25 are both powers of 5.

Therefore, the equation can be rewritten as 

$\displaystyle (5^3)^{6x-10}=(5^2)^{-x+5}$.

Using the Distributive Property, 

$\displaystyle 5^{18x-30}=5^{-2x+10}$. 

Since both sides now have the same base, set the two exponents equal to one another and solve:

$\displaystyle 18x-30=-2x+10$

Add 30 to both sides: $\displaystyle 18x=-2x+40$

Add $\displaystyle 2x$ to both sides: $\displaystyle 20x=40$

Divide both sides by 20: $\displaystyle x=2$

Both 27 and 9 are powers of 3, therefore the equation can be rewritten as 

$\displaystyle (3^2)^{4x+2}=(3^3)^{3x-4}$.

Using the Distributive Property, 

$\displaystyle 3^{8x+4}=3^{9x-12}$. 

Now that both sides have the same base, set the two exponenents equal and solve.

$\displaystyle 8x+4=9x-12$

Add 12 to both sides: $\displaystyle 8x+16=9x$

Subtract $\displaystyle 8x$ from both sides: $\displaystyle 16=x$

The first step in thist problem is divide both sides by three: $\displaystyle 2^{x}=8$. Then, recognize that 8 could be rewritten with a base of 2 as well ($\displaystyle 2^3$). Therefore, your answer is 3.

Let's convert $\displaystyle 64$ to base $\displaystyle 2$.

We know the following:

$\displaystyle 64=2*2*2*2*2*2$

Simplify.

$\displaystyle 2*2*2*2*2*2=2^6$

Solve.

 $\displaystyle x=6$

Let's convert $\displaystyle 2187$ to base $\displaystyle 3$.

We know the following:

$\displaystyle 2187=3*3*3*3*3*3*3$

Simplify.

$\displaystyle 3*3*3*3*3*3*3=3^7$

Solve.

$\displaystyle x=7$.