Algebra 2 PBL: Working With Logarithms

                                                     
The fundamental tool needed to appropriately deal with logarithms is the change of base formula. If an equation meets the constraints (a, b, and x are positive real numbers, and neither a nor b is one) then:

The change of base formula does exactly what it sounds like, it allows us to change the base of the logarithm so that we're able to obtain a solution. A good example may be log8^24 (8 being the base). If we were to evaluate that, no whole number "x" (8^x) could create 24. In that case, we can apply the change of base formula, so that it's expressed as log24/log8 which makes the solution approximately 1.53.

So essentially, something like log0.5^9 would be log9/log0.5, which would then equal -3.2.

Once you've gotten the change of base formula mastered, what comes next is easy. Something like log4^20=2x+5 is easy to solve, you just apply the change of base formula to the left side and then simplify from there like you would with a normal equation. 

log4^20=2x+5

log20/log4=2x+5

(log20/log4)-5=2x

(log20/log4-5)/2=x

x=-1.42

For graphing:

Seperate the two equations into y1 and y2, and do what's shown below.

You can also find a solution via graphing, as I show here: 

If you look at the place of intersection, it's at the same mark as x above, so it's verified!

To solve an equation with x on both sides, you do the same thing as you would above:

The solution to problem is approximately 1.87. Also, if there isn't a base for a logarithm such as with log x above, then it is generally accepted to be 10.

And last, but definitely not least, is solving equations with two logarithms! 

log5^(x-7)=-log4^x

The first step to solving what's above is writing a "system of equations", or y1=log5^(x-7) and y2= -log4^x. From there, you'll want to rewrite both equations via the change of base formula: logx-7/log5 and -logx/log4. After the first two steps, there's only graphing and identifying the solutions:

The solution for this system is approximately 7.1.

And that's all for this project! I realize that logarithms expand much farther than this, but for this project I  wanted to cover material mainly from the "Solving Logarithmic Equations Using Technology" time period.