{{ item.displayTitle }}

No history yet!

equalizer

rate_review

{{ r.avatar.letter }}

{{ u.avatar.letter }}

+

{{ item.displayTitle }}

{{ item.subject.displayTitle }}

{{ searchError }}

{{ courseTrack.displayTitle }} {{ statistics.percent }}% Sign in to view progress

{{ printedBook.courseTrack.name }} {{ printedBook.name }} {{ greeting }} {{userName}}

{{ 'ml-article-collection-banner-text' | message }}

{{ 'ml-article-collection-your-statistics' | message }}

{{ 'ml-article-collection-overall-progress' | message }}

{{ 'ml-article-collection-challenge' | message }} Coming Soon!

{{ 'ml-article-collection-randomly-challenge' | message }}

{{ 'ml-article-collection-community' | message }}

{{ 'ml-article-collection-follow-latest-updates' | message }}

Simplify the expression
using the properties of logarithms.

Show Solution

The Quotient Property of Logarithms states that the following is true for logarithms.
In our expression the first two terms and the last two terms both are a difference of two logarithms with the same base. We can use the Quotient Property of Logarithms to rewrite each pair into a logarithm of a quotient.
This expression is a sum of two logarithms with the same base. We can write them into a logarithm of a product using the Product Property of Logarithms.
Let's continue simplifying this expression.
Thus, the expression $g_{6}(8)−g_{6}(3)+g_{6}(9)−g_{6}(4)$ equals 1.

$g_{6}(38 ⋅49 )$

MultFrac

Multiply fractions

$g_{6}(3⋅48⋅9 )$

SimpQuot

Simplify quotient

$g_{6}(6)$

$g_{6}(6)=1$

1

The Change of Base Formula allows the logarithm of an arbitrary base to be rewritten as the quotient of two logarithms with another base.
With many calculators it is only possible to evaluate the common and the natural logarithm. The Change of Base Formula can then be used to evaluate logarithms of other bases.

$g_{c}(a)=g(c)g(a) andg_{c}(a)=ln(c)ln(a) $

An exponential equation can be solved by applying logarithms and using the Power Property of Logarithms. Consider the following equation.
### 1

### 2

### 3

After the power has been rewritten into a product, the unknown variable x can be isolated using inverse operations. Here, x gets isolated on the left-hand side when both sides of the equation are divided by $g(8).$
By using a calculator, an approximate value of x can be calculated. Here, $x≈0.53.$

Apply the logarithm

First, the equation is rewritten by applying a logarithm on both sides.

Rewrite using the Power Property of Logarithms

By using the Power Property of Logarithms, powers can be rewritten into a product.

Solve the resulting equation

Solve the equation
using the common logarithm. State the answer with three significant digits.

Show Solution

The given equation can be solved by applying the $g_{4}$ on both sides. Since many calculators are limited to only the common and the natural logarithms, it is often not possible to use $g_{4}.$ We can solve it anyway. Let's begin by applying the common logarithm on both sides.
The Power Property of Logarithms gives us the relationship $g_{b}(m_{n})=n⋅g_{b}(m).$ With this we can rewrite the left-hand side of the equation.
We can now solve the equation for x by dividing both sides of the equation with $g(4).$
The solution to the equation is $x≈2.32.$

$x⋅g(4)=g(25)$

DivEqn

$LHS/g(4)=RHS/g(4)$

$x=g(4)g(25) $

UseCalc

Use a calculator

$x=2.32192…$

RoundDec

Round to

$x≈2.32$

A logarithmic function $g(x)=g_{b}(x),$ is by definition the inverse of an exponential function $f(x)=b_{x}.$ This means that their function composition results in the identity function.

- $g(f(x))=g_{b}(b_{x})=x$
- $f(g(x))=b_{log_{b}(x)}=x$

The fact that a logarithm and a power with the same base undo

each other is what is known as the **inverse properties of logarithms**.

$g_{b}(b_{x})=xandb_{log_{b}(x)}=x $

They also hold true for the common logarithm and the natural logarithm.

$g(10_{x})=xandln(e_{x})=xand 10_{log(x)}=xe_{ln(x)}=x $

These properties together with other properties of logarithms permit to simplify logarithmic expressions and to solve equations involving logarithms and powers. Some particular examples are shown below.

Solve the equation
using the inverse properties of logarithms.

Show Solution

The inverse properties of logarithms tell us that a logarithm and a power with the same base undo each other.
To simplify the remaining two logarithms, we first need to rewrite their arguments.
Again, we can use the the inverse properties of logarithms to simplify the equation.
We now need to isolate x on the left-hand side to find the solution.
We have solved the equation, and its solution is $272 .$

$g_{b}(b_{x})=xandb_{log_{b}(x)}=x$

On the left-hand side, we can identify $ln$ as being the logarithm with base e and $g$ as the logarithm with base 10. Thus, we can simplify the factors $e_{ln(3_{4})}$ and $10_{log(x)}$ using the inverse properties of logarithms.
$33_{4}⋅x =2$

SimpQuot

Simplify quotient

33⋅x=2

DivEqn

$LHS/3_{3}=RHS/3_{3}$

$x=3_{3}2 $

CalcPow

Calculate power

$x=272 $

{{ 'mldesktop-placeholder-grade' | message }} {{ article.displayTitle }}!

{{ focusmode.exercise.exerciseName }}