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.

\(\displaystyle x=7\).