Properties of Logarithms

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From the course by University of California, Irvine

Pre-Calculus: Functions

311 ratings

University of California, Irvine

Pre-Calculus: Functions

311 ratings

This course covers mathematical topics in college algebra, with an emphasis on functions. The course is designed to help prepare students to enroll for a first semester course in single variable calculus. Upon completing this course, you will be able to: 1. Solve linear and quadratic equations 2. Solve some classes of rational and radical equations 3. Graph polynomial, rational, piece-wise, exponential and logarithmic functions 4. Find integer roots of polynomial equations 5. Solve exponential and logarithm equations 6. Understand the inverse relations between exponential and logarithm equations 7. Compute values of exponential and logarithm expressions using basic properties

From the lesson

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.

Introduction to Exponential & Logarithmic Functions 13:58

Solving Exponential Equations 3:56

Change of Base Formula for Logarithms 3:26

Converting Between Logarithmic and Exponential Equations 2:49

Properties of Logarithms 5:06

Meet the Instructors

Dr. Sarah Eichhorn
Associate Dean, Distance Learning & Tenured Lecturer
Dr. Rachel Cohen Lehman

[MUSIC] Let's work with the Properties of Logarithms.

For example, given that log x=3 and log y= -2, let's find the log of x to the

5th, y cubed. And the log of x squared divided by

square root of y. And we'll be using the following

properties of logs to help us. Where here a is any positive base not

equal to 1, M and N are positive numbers and P is any number.

So let's begin with the log of x to the 5th, y cubed.

Cubed. By this property down here, the logarithm

of a product is the sum of the logs, in other words this is equal to log of x ^ 5

+ log of y ^ 3. And by this last property down here, the

logarithm of a power, we can take this power and bring it down in front of the

logs. So let's do that to both of these

logarithm expressions here. We can the 5 down in front here, as well

as the 3 down in front here. Which gives us, 5 x log of x plus 3 x log

of Y. Now we're given that log x = 3, which we

can plug in here, as well as log y = -2, which we can plug in here.

Therefore, this is equal to 5*3, + 3*-2 or 15-6, which is 9.

Alright, and what about this last log we are asked to find? The log of x^2 divided

by the square root of y. Well, by the second property over here,

the logarithm of the quotient is equal to the difference in the logarithms.

That is, this is equal to log x ^ 2 - log of the square root of y.

Which is equal to log x ^ 2 and then minus log, and we can rewrite square root

of Y as Y of the 1/2 power. We wanted to write both of these

expressions with power's, so that we can bring these powers down in front of the

log, by this log rhythm with a power property down here.

That is, this is equal to 2 times log of X, minus 1 half, times log of 1.

Again we can use the fact that log X is equal to 3, and that log Y is equal to

negative 2. Which gives us 2 * 3 - 1/2 * -2 or 6 + 1

which = 7, which is our answer.

Alright, let's look at another example. [SOUND] Let's compute log base 2 of 24 -

log base 2 of 3. Now do we know a power that we can raise

2 to to get 24 or a power we can raise 2 to, To get 3.

We don't, so we could use a change of base formula

on each of these logs separately. But it's going to much easier if we use

these properties of logs. If we look here at this middle property,

we can use it in the reverse direction. In other words, we're starting with this,

and then we're going to condense it using this property.

That is, this is = log base 2 of 24 / 3, which = log base 2 of 8, which is = 3.

Since inverting to exponential form, we know that 2 cubed is equal to 8.

So this would be our answer. And this is how we work with properties

of logs. Thank you, and we'll see you next time.

[MUSIC]

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