# 4.5 Logarithmic properties

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In this section, you will:
• Use the product rule for logarithms.
• Use the quotient rule for logarithms.
• Use the power rule for logarithms.
• Expand logarithmic expressions.
• Condense logarithmic expressions.
• Use the change-of-base formula for logarithms.

In chemistry, pH is used as a measure of the acidity or alkalinity of a substance. The pH scale runs from 0 to 14. Substances with a pH less than 7 are considered acidic, and substances with a pH greater than 7 are said to be alkaline. Our bodies, for instance, must maintain a pH close to 7.35 in order for enzymes to work properly. To get a feel for what is acidic and what is alkaline, consider the following pH levels of some common substances:

• Battery acid: 0.8
• Stomach acid: 2.7
• Orange juice: 3.3
• Pure water: 7 (at 25° C)
• Human blood: 7.35
• Fresh coconut: 7.8
• Sodium hydroxide (lye): 14

To determine whether a solution is acidic or alkaline, we find its pH, which is a measure of the number of active positive hydrogen ions in the solution. The pH is defined by the following formula, where $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is the concentration of hydrogen ion in the solution

The equivalence of $\text{\hspace{0.17em}}-\mathrm{log}\left(\left[{H}^{+}\right]\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{log}\left(\frac{1}{\left[{H}^{+}\right]}\right)\text{\hspace{0.17em}}$ is one of the logarithm properties we will examine in this section.

## Using the product rule for logarithms

Recall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here. First, the following properties are easy to prove.

$\begin{array}{l}{\mathrm{log}}_{b}1=0\\ {\mathrm{log}}_{b}b=1\end{array}$

For example, $\text{\hspace{0.17em}}{\mathrm{log}}_{5}1=0\text{\hspace{0.17em}}$ since $\text{\hspace{0.17em}}{5}^{0}=1.\text{\hspace{0.17em}}$ And $\text{\hspace{0.17em}}{\mathrm{log}}_{5}5=1\text{\hspace{0.17em}}$ since $\text{\hspace{0.17em}}{5}^{1}=5.$

Next, we have the inverse property.

For example, to evaluate $\text{\hspace{0.17em}}\mathrm{log}\left(100\right),$ we can rewrite the logarithm as $\text{\hspace{0.17em}}{\mathrm{log}}_{10}\left({10}^{2}\right),$ and then apply the inverse property $\text{\hspace{0.17em}}{\mathrm{log}}_{b}\left({b}^{x}\right)=x\text{\hspace{0.17em}}$ to get $\text{\hspace{0.17em}}{\mathrm{log}}_{10}\left({10}^{2}\right)=2.$

To evaluate $\text{\hspace{0.17em}}{e}^{\mathrm{ln}\left(7\right)},$ we can rewrite the logarithm as $\text{\hspace{0.17em}}{e}^{{\mathrm{log}}_{e}7},$ and then apply the inverse property $\text{\hspace{0.17em}}{b}^{{\mathrm{log}}_{b}x}=x\text{\hspace{0.17em}}$ to get $\text{\hspace{0.17em}}{e}^{{\mathrm{log}}_{e}7}=7.$

Finally, we have the one-to-one property.

We can use the one-to-one property to solve the equation $\text{\hspace{0.17em}}{\mathrm{log}}_{3}\left(3x\right)={\mathrm{log}}_{3}\left(2x+5\right)\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ Since the bases are the same, we can apply the one-to-one property by setting the arguments equal and solving for $\text{\hspace{0.17em}}x:$

But what about the equation $\text{\hspace{0.17em}}{\mathrm{log}}_{3}\left(3x\right)+{\mathrm{log}}_{3}\left(2x+5\right)=2?\text{\hspace{0.17em}}$ The one-to-one property does not help us in this instance. Before we can solve an equation like this, we need a method for combining terms on the left side of the equation.

Recall that we use the product rule of exponents to combine the product of exponents by adding: $\text{\hspace{0.17em}}{x}^{a}{x}^{b}={x}^{a+b}.\text{\hspace{0.17em}}$ We have a similar property for logarithms, called the product rule for logarithms , which says that the logarithm of a product is equal to a sum of logarithms. Because logs are exponents, and we multiply like bases, we can add the exponents. We will use the inverse property to derive the product rule below.

Given any real number $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and positive real numbers and $\text{\hspace{0.17em}}b,$ where $\text{\hspace{0.17em}}b\ne 1,$ we will show

So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
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
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
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?