Properties of logarithms

There are several rules you’ll use often when working with logarithms. In this chapter, you’ll learn the main logarithm rules and how to apply them in basic algebraic equations.

Understanding logarithms

A logarithm tells you what exponent you need to raise a base to in order to get a given number. This idea becomes clearer once you start using logarithms in algebra.

For now, look at the parts of a logarithm using this example:

$lo g_{2} (8) = 3$

The $2$ is the base. It’s the smaller number written as a subscript next to $lo g$ .
The $8$ is the number the logarithm is applied to (the input).
The $3$ is the output. It means that $2$ must be raised to the power $3$ to get $8$ .

In other words, $lo g_{2} (8) = 3$ is equivalent to $2^{3} = 8$ .

Logarithm rules using parentheses

Multiplication

Multiplication inside a logarithm can be rewritten as a sum of logarithms. This is useful because it breaks one complicated log into simpler pieces.

Start with:

$lo g_{2} (8 * x)$

Rewrite it as:

$lo g_{2} (8) + lo g_{2} (x)$

Notice that both logarithms keep the same base, $2$ . When you expand a logarithm this way, every term must have the same base.

Division

Division inside a logarithm can be rewritten as a difference of logarithms. The base stays the same.

For example:

$lo g_{2} (\frac{8}{x}) = lo g_{2} (8) - lo g_{2} (x)$

The numerator becomes the first logarithm, and the denominator becomes the logarithm that is subtracted.

Exponents

If the expression inside the logarithm has an exponent, you can move that exponent to the front as a multiplier.

Example:

$lo g_{2} (x^{8}) = 8 * lo g_{2} (x)$

The base and the inside expression stay the same; only the exponent moves outside.

Reverse rules

The three rules above work in both directions. You can expand a single logarithm into multiple logs, and you can also combine multiple logs back into one.

Rewrite the following expression as a single logarithm

$lo g_{10} (8) + lo g_{10} (7) + lo g_{10} (x)$

This is a sum of logarithms with the same base, so it comes from the multiplication rule. Combine the terms by multiplying inside one logarithm:

$lo g_{10} (8 * 7 * x) = lo g_{10} (56 x)$

Simplify the following logarithmic expression:

$3 * lo g_{6} (x)$

(spoiler)

Use the exponent rule in reverse. The exponent rule says you can pull an exponent out of a logarithm; reversing it means you put the multiplier back as an exponent.

Thus, our final answer is:

$lo g_{6} (x^{3})$

Change of base

Sometimes you need to evaluate a logarithm with a base other than $10$ . Some calculators (like the TI-84) can compute logs with other bases directly, but others cannot. Here are both approaches.

Logarithms in a graphing calculator

On a TI-84, you can evaluate a logarithm by pressing the $lo g$ button, entering the number inside the log, then a comma, then the base.

In calculator notation, this looks like $lo g (number, base)$ .

For example, to find $lo g_{2} (30)$ , type (without quotation marks) “ $lo g (2, 30)$ .” This comes out to equal $4.9$ .

Change of base formula

If your calculator only has $lo g$ (base $10$ ) or $ln$ (base $e$ ), you can rewrite a logarithm using the change of base formula:

$lo g_{b} (a) = \frac{lo g _{x} ( a )}{lo g _{x} ( b )}$

Here, the base $x$ in the fraction can be any number greater than $1$ . In practice, you usually choose $10$ so you can use the standard $lo g$ button.

That gives:

$lo g_{b} (a) = \frac{lo g _{10} ( a )}{lo g _{10} ( b )}$

Now evaluate $lo g_{2} (30)$ :

$lo g_{2} (30) = \frac{lo g _{10} ( 30 )}{lo g _{10} ( 2 )} = 4.9$

Logarithms in algebra

Sometimes you’ll be asked to solve for $x$ in a logarithmic equation, such as:

$lo g_{x} (9) = 2$

A key idea is that you can rewrite a logarithmic equation in exponential form. In general,

$lo g_{a} (b) = c means b = a^{c} .$

Apply that here:

$lo g_{x} (9) 993 = 2 = x^{2} = x = x$

One way to remember this is to “swap” the base and the exponent:

The base (the small subscript) becomes the base of an exponential expression.
The number on the right side becomes the exponent.

Examples

What is the value of $x$ ?

$lo g_{2} (x) = 4$

(spoiler)

Rewrite in exponential form by moving the base ( $2$ ) to the right side and using $4$ as the exponent:

$lo g_{2} (x) x x = 4 = 2^{4} = 16$

Let’s try another!

Find the value of $x$ for the following equation:

$lo g_{4} (64) = x$

(spoiler)

Rewrite in exponential form:

$64 = 4^{x}$

Since $4^{3} = 64$ :

$x = 3$

Using logarithms to solve for $x$ as an exponent

Sometimes $x$ appears as an exponent, and you want to “bring it down” so you can solve for it. You can do this by taking the logarithm of both sides.

Find the value of $x$ for the following equation in terms of a logarithm:

$33 = 2^{x}$

Take the log of both sides using base $2$ :

$lo g_{2} (33) = lo g_{2} (2^{x})$

Now use the exponent rule to simplify the right side:

$lo g_{2} (33) = x$

The key point is that logarithms let you rewrite an exponential equation so the exponent becomes a regular number you can solve for.

Natural logarithms and the number e

Occasionally, you’ll see a logarithm called a natural logarithm, which is a logarithm with base $e$ :

$ln (x) = l o g_{e} (x)$

The number $e$ is approximately:

$e \approx 2.71828$

The constant $e$ is an irrational number like $π$ . It never ends or repeats. It appears in many real-life situations such as population growth, compound interest, and other processes that model continuous growth. That’s why logarithms with base $e$ are called “natural.”

$y = e^{x}$

Then

$x = l n (y)$

Natural logarithms are the inverse of exponential functions with a base of $e$ . The most important rule to remember is that the natural log of $e$ is equal to 1.

$ln (e) = 1$

For example, here are some common ways to convert between exponential form and natural logarithm form.

Exponential Form	Log Form
$e^{3}$	$ln (20.085) = 3$
$e^{0}$	$ln (1) = 0$
$e^{-} 1$	$ln (\frac{1}{e}) = - 1$

To solve an equation with a $e$ in it, you can “take the natural log” of both sides of the equation. Here’s an example:

Examples

What is the value of $x$ ?

$5 e^{3} x = 40$

(spoiler)

Step 1: Divide both sides by 5

$e^{3} x = 8$

Step 2: Take the natural log of both sides

$ln (e^{3} x) = ln (8)$

Step 3: Bring the exponent down

$3 x \times ln (e) = ln (8)$

Step 4: Since \ln(e) = 1, we can simplify

$3 x = ln (8)$

Step 5: Finish solving

$x = \frac{ln ( 8 )}{3}$

Key points

Multiplication. Take each term being multiplied and put them into logarithms of the same base as a total sum: $lo g_{a} (b * c) = lo g_{a} (b) + lo g_{a} (c)$

Division. Take each term being divided and put them into logarithms of the same base as a total difference: $lo g_{a} (b / c) = lo g_{a} (b) - lo g_{a} (c)$

Exponent. Take the exponent from inside the logarithm and multiply it in front of the logarithm: $lo g_{a} (b^{c}) = c * lo g_{a} (b)$

Algebra. To use logarithms in algebra, move the base to the right side of the equation, raising what used to be there as the power of the base: $lo g_{a} (b) = c$ is simplified as $b = a^{c}$ . Lower $x$ from an exponent by taking the log of both sides of the equation.

Change of base. Change the base of a logarithm to $10$ in order to put it into a basic calculator by using the formula $lo g_{b} (a) = \frac{l o g _{10} ( a )}{l o g _{10} ( b )}$ . Alternatively, calculate a logarithm with an abnormal base in your calculator by typing “ $lo g (number, base)$ ”.

Natural logarithms. Natural logarithms are just logarithms with a base of $e$ . Remember that $ln (e) = 1$ and that you can take the natural log of both sides of an equation in order to “cancel out” the $e$ .

There are several rules you’ll use often when working with logarithms. In this chapter, you’ll learn the main logarithm rules and how to apply them in basic algebraic equations.

Understanding logarithms

A logarithm tells you what exponent you need to raise a base to in order to get a given number. This idea becomes clearer once you start using logarithms in algebra.

For now, look at the parts of a logarithm using this example:

$lo g_{2} (8) = 3$

The $2$ is the base. It’s the smaller number written as a subscript next to $lo g$ .
The $8$ is the number the logarithm is applied to (the input).
The $3$ is the output. It means that $2$ must be raised to the power $3$ to get $8$ .

In other words, $lo g_{2} (8) = 3$ is equivalent to $2^{3} = 8$ .

Logarithm rules using parentheses

Multiplication

Multiplication inside a logarithm can be rewritten as a sum of logarithms. This is useful because it breaks one complicated log into simpler pieces.

Start with:

$lo g_{2} (8 * x)$

Rewrite it as:

$lo g_{2} (8) + lo g_{2} (x)$

Notice that both logarithms keep the same base, $2$ . When you expand a logarithm this way, every term must have the same base.

Division

Division inside a logarithm can be rewritten as a difference of logarithms. The base stays the same.

For example:

$lo g_{2} (\frac{8}{x}) = lo g_{2} (8) - lo g_{2} (x)$

The numerator becomes the first logarithm, and the denominator becomes the logarithm that is subtracted.

Exponents

If the expression inside the logarithm has an exponent, you can move that exponent to the front as a multiplier.

Example:

$lo g_{2} (x^{8}) = 8 * lo g_{2} (x)$

The base and the inside expression stay the same; only the exponent moves outside.

Reverse rules

The three rules above work in both directions. You can expand a single logarithm into multiple logs, and you can also combine multiple logs back into one.

Rewrite the following expression as a single logarithm

$lo g_{10} (8) + lo g_{10} (7) + lo g_{10} (x)$

This is a sum of logarithms with the same base, so it comes from the multiplication rule. Combine the terms by multiplying inside one logarithm:

$lo g_{10} (8 * 7 * x) = lo g_{10} (56 x)$

Simplify the following logarithmic expression:

$3 * lo g_{6} (x)$

(spoiler)

Use the exponent rule in reverse. The exponent rule says you can pull an exponent out of a logarithm; reversing it means you put the multiplier back as an exponent.

Thus, our final answer is:

$lo g_{6} (x^{3})$

Change of base

Sometimes you need to evaluate a logarithm with a base other than $10$ . Some calculators (like the TI-84) can compute logs with other bases directly, but others cannot. Here are both approaches.

Logarithms in a graphing calculator

On a TI-84, you can evaluate a logarithm by pressing the $lo g$ button, entering the number inside the log, then a comma, then the base.

In calculator notation, this looks like $lo g (number, base)$ .

For example, to find $lo g_{2} (30)$ , type (without quotation marks) “ $lo g (2, 30)$ .” This comes out to equal $4.9$ .

Change of base formula

If your calculator only has $lo g$ (base $10$ ) or $ln$ (base $e$ ), you can rewrite a logarithm using the change of base formula:

$lo g_{b} (a) = \frac{lo g _{x} ( a )}{lo g _{x} ( b )}$

Here, the base $x$ in the fraction can be any number greater than $1$ . In practice, you usually choose $10$ so you can use the standard $lo g$ button.

That gives:

$lo g_{b} (a) = \frac{lo g _{10} ( a )}{lo g _{10} ( b )}$

Now evaluate $lo g_{2} (30)$ :

$lo g_{2} (30) = \frac{lo g _{10} ( 30 )}{lo g _{10} ( 2 )} = 4.9$

Logarithms in algebra

Sometimes you’ll be asked to solve for $x$ in a logarithmic equation, such as:

$lo g_{x} (9) = 2$

A key idea is that you can rewrite a logarithmic equation in exponential form. In general,

$lo g_{a} (b) = c means b = a^{c} .$

Apply that here:

$lo g_{x} (9) 993 = 2 = x^{2} = x = x$

One way to remember this is to “swap” the base and the exponent:

The base (the small subscript) becomes the base of an exponential expression.
The number on the right side becomes the exponent.

Examples

What is the value of $x$ ?

$lo g_{2} (x) = 4$

(spoiler)

Rewrite in exponential form by moving the base ( $2$ ) to the right side and using $4$ as the exponent:

$lo g_{2} (x) x x = 4 = 2^{4} = 16$

Let’s try another!

Find the value of $x$ for the following equation:

$lo g_{4} (64) = x$

(spoiler)

Rewrite in exponential form:

$64 = 4^{x}$

Since $4^{3} = 64$ :

$x = 3$

Using logarithms to solve for $x$ as an exponent

Sometimes $x$ appears as an exponent, and you want to “bring it down” so you can solve for it. You can do this by taking the logarithm of both sides.

Find the value of $x$ for the following equation in terms of a logarithm:

$33 = 2^{x}$

Take the log of both sides using base $2$ :

$lo g_{2} (33) = lo g_{2} (2^{x})$

Now use the exponent rule to simplify the right side:

$lo g_{2} (33) = x$

The key point is that logarithms let you rewrite an exponential equation so the exponent becomes a regular number you can solve for.

Natural logarithms and the number e

Occasionally, you’ll see a logarithm called a natural logarithm, which is a logarithm with base $e$ :

$ln (x) = l o g_{e} (x)$

The number $e$ is approximately:

$e \approx 2.71828$

$y = e^{x}$

Then

$x = l n (y)$

Natural logarithms are the inverse of exponential functions with a base of $e$ . The most important rule to remember is that the natural log of $e$ is equal to 1.

$ln (e) = 1$

For example, here are some common ways to convert between exponential form and natural logarithm form.

Exponential Form	Log Form
$e^{3}$	$ln (20.085) = 3$
$e^{0}$	$ln (1) = 0$
$e^{-} 1$	$ln (\frac{1}{e}) = - 1$

To solve an equation with a $e$ in it, you can “take the natural log” of both sides of the equation. Here’s an example:

Examples

What is the value of $x$ ?

$5 e^{3} x = 40$

(spoiler)

Step 1: Divide both sides by 5

$e^{3} x = 8$

Step 2: Take the natural log of both sides

$ln (e^{3} x) = ln (8)$

Step 3: Bring the exponent down

$3 x \times ln (e) = ln (8)$

Step 4: Since \ln(e) = 1, we can simplify

$3 x = ln (8)$

Step 5: Finish solving

$x = \frac{ln ( 8 )}{3}$

Key points

Multiplication. Take each term being multiplied and put them into logarithms of the same base as a total sum: $lo g_{a} (b * c) = lo g_{a} (b) + lo g_{a} (c)$

Division. Take each term being divided and put them into logarithms of the same base as a total difference: $lo g_{a} (b / c) = lo g_{a} (b) - lo g_{a} (c)$

Exponent. Take the exponent from inside the logarithm and multiply it in front of the logarithm: $lo g_{a} (b^{c}) = c * lo g_{a} (b)$

Understanding logarithms

Logarithm rules using parentheses

Multiplication

Division

Exponents

Reverse rules

Change of base

Logarithms in a graphing calculator

Change of base formula

Logarithms in algebra

Examples

Using logarithms to solve for x as an exponent

Natural logarithms and the number e

Examples

Properties of logarithms

Understanding logarithms

Logarithm rules using parentheses

Multiplication

Division

Exponents

Reverse rules

Change of base

Logarithms in a graphing calculator

Change of base formula

Logarithms in algebra

Examples

Using logarithms to solve for x as an exponent

Natural logarithms and the number e

Examples

Using logarithms to solve for $x$ as an exponent

Using logarithms to solve for $x$ as an exponent