# 4.3 Logarithmic functions

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In this section, you will:
• Convert from logarithmic to exponential form.
• Convert from exponential to logarithmic form.
• Evaluate logarithms.
• Use common logarithms.
• Use natural logarithms.

In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homes http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/#summary. Accessed 3/4/2013. . One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings, http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#summary. Accessed 3/4/2013. like those shown in [link] . Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a 7.0 on the Richter Scale http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/. Accessed 3/4/2013. whereas the Japanese earthquake registered a 9.0. http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#details. Accessed 3/4/2013.

The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is ${10}^{8-4}={10}^{4}=10,000$ times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten function upon which it depends.

## Converting from logarithmic to exponential form

In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is $\text{\hspace{0.17em}}{10}^{x}=500,$ where $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ represents the difference in magnitudes on the Richter Scale . How would we solve for $\text{\hspace{0.17em}}x?$

We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve $\text{\hspace{0.17em}}{10}^{x}=500.\text{\hspace{0.17em}}$ We know that $\text{\hspace{0.17em}}{10}^{2}=100\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{10}^{3}=1000,$ so it is clear that $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ must be some value between 2 and 3, since $\text{\hspace{0.17em}}y={10}^{x}\text{\hspace{0.17em}}$ is increasing. We can examine a graph, as in [link] , to better estimate the solution.

Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in [link] passes the horizontal line test. The exponential function $\text{\hspace{0.17em}}y={b}^{x}\text{\hspace{0.17em}}$ is one-to-one , so its inverse, $\text{\hspace{0.17em}}x={b}^{y}\text{\hspace{0.17em}}$ is also a function. As is the case with all inverse functions, we simply interchange $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ and solve for $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ to find the inverse function. To represent $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ as a function of $\text{\hspace{0.17em}}x,$ we use a logarithmic function of the form $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right).\text{\hspace{0.17em}}$ The base $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ logarithm of a number is the exponent by which we must raise $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ to get that number.

We read a logarithmic expression as, “The logarithm with base $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is equal to $\text{\hspace{0.17em}}y,$ ” or, simplified, “log base $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}y.$ ” We can also say, “ $b\text{\hspace{0.17em}}$ raised to the power of $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}x,$ ” because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since $\text{\hspace{0.17em}}{2}^{5}=32,$ we can write $\text{\hspace{0.17em}}{\mathrm{log}}_{2}32=5.\text{\hspace{0.17em}}$ We read this as “log base 2 of 32 is 5.”

difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott