# 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

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
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas