Logarithm Rules | Log Rules | ln Rules – Definition and laws

Here are all the Logarithm formulas you need in order to succeed in the course. Mastering these formulas will save you a lot of time in solving exercises and will help you avoid unnecessary calculation errors

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Definition

$\log_a x = b \Longleftrightarrow a^b=x$

for

$a>0 \text{ and } a\neq 1$

From the definition we get

$x>0$

Example

$\log_2 8 = 3$

Since

$2^3 = 8$

Properties:

1. Identity Rule

$\log_a a=1$

for any a.

Example

$\log_5 5=1$

2. Zero Rule

$\log_a 1=0$

for all a.

Example

$\log_3 1=0$

Logarithm Rules:

• Product Rule
• Quotient Rule
• Power Rule
• Exponent Rule
• Changing Log Base
• Using ln and base 10
• Log in Inequality

Product Rule

This rule allows you to split a log with multiplication in it into two logs:

$\log_a (x\cdot y)=\log_a x +\log_a y$

You can use the formula in both directions, that is, if there is a multiplicity within the log, then it can be split into a sum of two logs and if there is a sum of two logs with the same base, then it can be converted into one log of multiples.

However, if there is a sum within the log, then it cannot be split:

$\log_a (x+y)\neq\log_a x \cdot\log_a y$

Example

$\log_2 8\cdot 4=\log_2 8 +\log_2 4=3+2=5$

Quotient Rule

This rule allows you to split a log with division in it into two logs:

$\log_a \frac{x}{y}=\log_a x -\log_a y$

The formula can be used in both directions, that is, if there is a difference within the log, then it can be split into two logs and if there is a difference of two logs with the same base, then they can be converted into one log of division.

However, if there is a difference within the log, then it cannot be split:

$\log_a (x-y)\neq\frac{\log_a x}{\log_a y}$

Example

$\log_3 \frac{27}{9}=\log_3 27 -\log_3 9=3-2=1$

By property 1 we have

$\log_3 \frac{27}{9}=\log_3 3=1$

Power Rule

This rule allows for flexibility in the position of the coefficient in the log:

$\log_a (x^b)=b\log_a x$

The law basically means that the coefficient can remain as a coefficient (multiplied by the log) or move to the power of the expression within the log. You can play with it and move the coefficient according to the exercise.

Example

$\log_5 625=\log_5 (5^4)=4\log_5 5=4\cdot 1=4$

Exponent Rule

$a^{\log_a x}=x$

This is the only rule that let you get rid of the log in the power of expression. Therefore, when there is a log in the power of the expression, there is no doubt on which formula to use

Example

$2^{\log_2 7}=7$

Changing Log Base

This rule allows to change the log base:

$\log_a x=\frac{\log_b x}{\log_b a}$

When you have logs in different bases, you need to choose the base that is most convenient – usually, this will be the smallest factor that divides all bases in the exercise – and then change all the logs in the exercise to the chosen base according to the formula above.

Example

$\log_4 8=\frac{\log_2 8}{\log_2 4}=\frac{3}{2}$

Using ln and base 10

Two unique log bases worth knowing are

1. ln is the base e logarithm, meaning:

$\log_e x=\ln x$

e is a mathematical irrational number, also called Euler’s number. The first digits are

$e=2.718...$

Note: when there is an ln function, all logarithm rules can and should be used.

2. When no base appears in the log, it means it is the 10 base log. That is,

$\log x=\log_{10} x$

Note: this is the base used in calculators.

Log in Inequality

We will use the fact that an exponential function is a monotonic function that increases when its basis is greater than one, so if x> y then

$a^x>a^y, a>1$

Hence, if

$\log_a x> b$

Then also

$a^{\log_a x}>a^b$

And this equals the inequality

$x>a^b$

And we got

$\log_a x> b, a>1\Longrightarrow x > a^b$

Note that when the log base (a) is between zero and one, the exponential function is a decreasing monotonic function, and we obtain:

$\log_a x> b, 0<a<1\Longrightarrow x < a^b$

Note: Today there are online logarithm calculators that evaluate logarithm exercises correctly and quickly. It even shows the steps of the solution. Although it cannot be used in the exam, it is very effective in practicing the material. Use one to check your final answers, when you are not sure.

Press here for exercises and solutions in logarithms