# 6.5 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. The pH of hydrochloric acid is tested with litmus paper. (credit: David Berardan)

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

exercise 1.2 solution b....isnt it lacking
I dnt get dis work well
what is one-to-one function
what is the procedure in solving quadratic equetion at least 6?
Almighty formula or by factorization...or by graphical analysis
Damian
I need to learn this trigonometry from A level.. can anyone help here?
yes am hia
Miiro
tanh2x =2tanhx/1+tanh^2x
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)=cotb ... pls some one should help me with this..thanks in anticipation
f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)= cotb
favour
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
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Amit
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Dorbor
well
Biswajit
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Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
helo
Akash
hlo
Akash
Hello
Hudheifa
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
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