High School Math Solutions – Exponential Equation Calculator

Solving exponential equations is pretty straightforward; there are basically two techniques:

• If the exponents on both sides of the equation have the same base, you can use the fact that: If a^x=a^y   then x=y.  Now simply solve the equation.

• In all other cases, take the log of both sides (this might require some manipulation) and solve for the variable.  The logarithm property ln(a^x)=xln(a) makes this a fairly simple task.

Let’s start with an example of exponents with the same base (click here):

Here’s another example that requires some manipulation to get the same base on both sides (click here):

Here’s an example using the logarithm properties (click here):

Cheers,
Michal

High School Math Solutions – Logarithmic Equation Calculator

Logarithmic equations are equations involving logarithms. In this segment we will cover equations with logarithms with the same base; logarithms with different base are much harder.

To solve logarithmic equations first manipulate the equation to have one log with the same base on each side so you can drop the logs (the only way two logs can be equal is if their arguments are equal). Then solve the linear or polynomial equation (we know how to do that).  The tricky part is that you have to check the solutions by plugging them into the original logarithmic equation (you can’t take logs of negative numbers).

Some useful logarithm properties to get familiar with before we start:

• n=log_a(aⁿ)
• log_a(x)+ log_a(y)= log_a(xy)
• nlog_a(x)= log_a(xⁿ)
• log_a(x)-log_a(y)= log_a(x/y)

Here’s an example (click here):

Here’s a more advanced example (click here):

Here’s an example with natural logarithms (click here):

Cheers,
Michal

Spinning The Unit Circle (Evaluating Trig Functions )

If you’ve ever taken a ferris wheel ride then you know about periodic motion, you go up and down over and over again.  The height of the seat is a periodic function of time; it rises and falls in a smooth, repeating manner.   Trigonometric functions can be defined in terms of the unit circle, the circle of radius one centered at the origin.  Sine and cosine are periodic functions with period 2π.  For angles greater than 2π or less than −2π, simply continue to rotate around the circle, just like the ferris wheel.

Evaluating trigonometric functions is all about knowing the unit circle (It is strongly recommended to memorize the trivial angles).

Let’s see how it works, here’s an example using a simple manipulation (click here):

Here’s an example using the periodic property (click here):

Here’s an example using trigonometric identity (click here):

Evaluating trigonometric equations might be somewhat tricky at first, but with practice you will use the same techniques over and over again…

Cheers,
Michal