	Algebra Tutorials!

	Thursday 21st of November



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	Multiplying Two Fractions Whose Numerators Are Both 1
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	Building Up the Denominator
	Adding and Subtracting Decimals
	Solving Quadratic Equations
	Scientific Notation
	Like Radical Terms
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	Subtracting Reverses
	Solving Linear Equations
	Dividing Rational Expressions
	Complex Numbers
	Solving Linear Inequalities
	Working with Fractions
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	Simplifying Expressions That Contain Negative Exponents
	Rationalizing the Denominator
	Decimals
	Estimating Sums and Differences of Mixed Numbers
	Algebraic Fractions
	Simplifying Rational Expressions
	Linear Equations
	Dividing Complex Numbers
	Simplifying Square Roots That Contain Variables
	Simplifying Radicals Involving Variables
	Compound Inequalities
	Factoring Special Quadratic Polynomials
	Simplifying Complex Fractions
	Rules for Exponents
	Finding Logarithms
	Multiplying Polynomials
	Using Coordinates to Find Slope
	Variables and Expressions
	Dividing Radicals
	Using Proportions and Cross
	Solving Equations with Radicals and Exponents
	Natural Logs
	The Addition Method
	Equations

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Natural Logs

Notation for Natural Logs

A natural logarithm is a log with base e. The number e is an irrational number whose value is approximately 2.718.

Natural logs are usually written with the abbreviation ln rather than log_e.

For example:

log_e57 = ln 57

log_e4.92 = ln 4.92

log_e 0.56 = ln 0.56

Note:

The abbreviation â€œlnâ€ comes from the French term for natural log, â€œlogarithme nÃ©pÃ©rien.â€

Definition â€” Natural Logarithm

A natural log is a log whose base is e. It is written like this:

ln x = log_ex

Here, x > 0.

Note:

The number e occurs naturally in applications such as compound interest, population growth, and radioactive decay.

Finding Natural Logs

We can find the value of some natural logs by switching from logarithmic notation to exponential notation.

Example 1

Find: ln e³

Solution

The base of a natural log is e.

Therefore, write ln e³ as log_ee³.

Set x equal to loge e³.

Rewrite in exponential form.

Use the Principle of Exponential Equality to solve for x.

So, ln e³ = 3.

x =

e^x =

x =

log_ee³

e³

In the last example we did not need a calculator because the argument was a power of e. However, in many cases we will need a calculator to find the natural log of a number.

Example 2

Solve: ln x = 7.92. Round to the nearest whole number.

Solution

Write ln x as log_ex.

Rewrite in exponential form.

Use your calculator to compute e^7.92.

Then round to the nearest whole number.

So, ln 2752 ≈ 7.92.

ln x

log_e x

e^7.92

2752

= 7.92

= x

≈ x

Example 3

Solve: ln x = -2.53

Solution

Write ln x as logex.

Rewrite in exponential form.

Use your calculator to compute e^-2.53.

Then round to two decimal places.

So, ln 0.08 ≈ -2.53.

ln x

log_e x

e^-2.53

0.08

= -2.53

= x

≈ x

Note:

Remember, the number e^-2.53 is a positive number, even though the exponent is negative.

To show this, we can write: