Solving Exponential Equations Algebraically Isolate the exponential expression on one side. Take the logarithm of both sides. The base for the logarithm should be the same as the base in the exponential expression.

In addition, each spiral originates from the same point. Limacons The next type of interesting graphs that we will explore are limacons.

Equations for limacons have the form where a and b are nonzero real numbers. There are four basic shapes. The following graphs show examples of each type of limacon. Limacons with an Inner Loop blue pink Notice the change from cos to sin rotates the limacon 90 degrees in the clockwise direction.

What if we change the positive sign to a negative sign and make a smaller? The other change, making a smaller, creates a larger inner loop.

Lastly, we want to explore what happens as b grows larger. The graphs for the following polar equations are shown. Furthermore, the reverse holds. As b decreases, the limacon gets smaller. Dimpled Limacons Several examples of dimpled limacons are shown along with their associated polar equations.

The change from positive to negative has the same effect as in the previous graphs. Convex Limacons The following polar equations and their graphs are examples of convex limacons.

Cardioids Cardioids are the fourth type of limacon.

The following graphs are several examples of cardiods. This is true for the other types of limacons. Hence, changing a to its opposite has no effect on the graph of a limacon.

For example, the following polar equations are reflections or mirror images of each other. Try graphing these polar equations to test our assumption. Similarly, as b decreases the size of the cardioid gets smaller. Pretty Petals Next we want to explore polar equations that produce graphs similar to that of roses.

What is the effect of a and n on the graph? Consider several more examples. First, a determines the length of each petal. What does the 8 in the equation do? By now, you should realize that the number of leaves is determined by n. From our previous examples, the number of leaves is twice n when n is even.

Does this always hold true? Try these other examples to investigate this assumption.

We can make a new generalization, however, that does hold true. There is a difference between the number of rose petals when n is odd and even. When n is odd, the number of petals is n. When n is even, the number of rose petals is 2n.

What if we change cos to sin? Does the polar graph still represent a rose curve? What effect does this change have on the polar graph?Aug 23, · Edit Article How to Solve Logarithms. In this Article: Solve for X Solve for X Using the Logarithmic Product Rule[3] Solve for X Using the Logarithmic Quotient Rule Community Q&A Logarithms might be intimidating, but solving a logarithm is much simpler once you realize that logarithms are just another way to write out exponential pfmlures.com: 28K.

Algebra 2 Here is a list of all of the skills students learn in Algebra 2! These skills are organized into categories, and you can move your mouse over any skill name to preview the skill. Exponential and Logarithmic Equations.

Exponential and logarithmic functions are two important functions that are useful in applications of calculus. We will discuss properties and graphs of these special functions. We will also learn to solve equations involving exponential and logarithmic functions.

And now, writing this in the equivalent. log a a x = x; log 10 x = x; ln e x = x; a log a x = x; 10 log x = x; e ln x = x; Solving Exponential Equations Algebraically. Isolate the exponential expression on one side. Take the logarithm of both sides.

The base for the logarithm should be the same as the base in the exponential expression. Writing Exponential and Logarithmic Equations from a Graph Writing Exponential Equations from Points and Graphs. You may be asked to write exponential equations, such as the following.

Solution We can solve both of these equations by translating from exponential form to logarithmic form. (a) Write the given equation in logarithmic form: $4^{-x^2} = 1/64$.

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Logarithmic Equations: Natural Base - Simple Equations