# 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.”

#### Questions & Answers

how to understand calculus?
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
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i want to sure my answer of the exercise
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how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
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meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
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32.243
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It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
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